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<p>Last updated on<strong>October 23, 2025</strong></p>
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<p>Last updated on<strong>October 23, 2025</strong></p>
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<p>If A is a square matrix, then its transpose (Aᵀ) is obtained by interchanging the rows with columns. When the original matrix A is multiplied by its inverse (A⁻¹), it gives the identity matrix (I). Mathematically, this is expressed as A · A⁻¹ = I. In this article, we will look at how transpose affects the orthogonality of a matrix.</p>
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<p>If A is a square matrix, then its transpose (Aᵀ) is obtained by interchanging the rows with columns. When the original matrix A is multiplied by its inverse (A⁻¹), it gives the identity matrix (I). Mathematically, this is expressed as A · A⁻¹ = I. In this article, we will look at how transpose affects the orthogonality of a matrix.</p>
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<h2>What is Orthogonal Matrix?</h2>
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<h2>What is Orthogonal Matrix?</h2>
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<p>What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math</p>
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<p>What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math</p>
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<p>▶</p>
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<p>▶</p>
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<p>If a<a>square</a>matrix transpose is equal to its<a>inverse</a>, then it is known as an orthogonal matrix, where \(A^{T} = A^{-1}\). We can use this definition to derive an important property of orthogonal matrices.</p>
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<p>If a<a>square</a>matrix transpose is equal to its<a>inverse</a>, then it is known as an orthogonal matrix, where \(A^{T} = A^{-1}\). We can use this definition to derive an important property of orthogonal matrices.</p>
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<p><strong>Proof:</strong></p>
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<p><strong>Proof:</strong></p>
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<p><strong>Given</strong>: \(A^{T} = A^{-1}\)</p>
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<p><strong>Given</strong>: \(A^{T} = A^{-1}\)</p>
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<p>Multiply both sides by A: \(A^{T} = A^{-1}\)</p>
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<p>Multiply both sides by A: \(A^{T} = A^{-1}\)</p>
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<p>Since \(AA^{-1} = I\), where I is the<a></a><a>identity matrix</a>, \(AA^{T} = I\) </p>
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<p>Since \(AA^{-1} = I\), where I is the<a></a><a>identity matrix</a>, \(AA^{T} = I\) </p>
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<p>On<a>multiplying</a>both sides of the original<a>equation</a>by A, we get \(A^TA = A^{-1}A = I\)</p>
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<p>On<a>multiplying</a>both sides of the original<a>equation</a>by A, we get \(A^TA = A^{-1}A = I\)</p>
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<p>So, \(AA^T = A^TA = I\)</p>
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<p>So, \(AA^T = A^TA = I\)</p>
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<p>This means that a matrix A is orthogonal if the<a>product</a>of the matrix and its transpose results in the identity matrix. This shows that a matrix can only be orthogonal if it produces an identity matrix when multiplied by its transpose.</p>
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<p>This means that a matrix A is orthogonal if the<a>product</a>of the matrix and its transpose results in the identity matrix. This shows that a matrix can only be orthogonal if it produces an identity matrix when multiplied by its transpose.</p>
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<ol></ol><h2>Properties of an Orthogonal Matrix</h2>
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<ol></ol><h2>Properties of an Orthogonal Matrix</h2>
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<p>Orthogonal matrices have structural and algebraic properties that define their characteristics. Some important properties of orthogonal matrices are listed below:</p>
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<p>Orthogonal matrices have structural and algebraic properties that define their characteristics. Some important properties of orthogonal matrices are listed below:</p>
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<ol><li>Inverse and transpose of the matrix are equal, i.e., \(A^{-1} = A^T\). </li>
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<ol><li>Inverse and transpose of the matrix are equal, i.e., \(A^{-1} = A^T\). </li>
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<li>The identity matrix is the product of the orthogonal matrix and its transpose, i.e., \(AA^T = A^TA = I\). </li>
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<li>The identity matrix is the product of the orthogonal matrix and its transpose, i.e., \(AA^T = A^TA = I\). </li>
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<li>Orthogonal matrices are always non-singular, with<a>determinant</a>\(det(A) = ±1\). (Note: Determinant can also be -1 for reflection matrices). </li>
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<li>Orthogonal matrices are always non-singular, with<a>determinant</a>\(det(A) = ±1\). (Note: Determinant can also be -1 for reflection matrices). </li>
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<li>An orthogonal matrix is diagonal only if the diagonal entries are either 1 or -1, and off-diagonal entries are zero. </li>
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<li>An orthogonal matrix is diagonal only if the diagonal entries are either 1 or -1, and off-diagonal entries are zero. </li>
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<li>Since the transpose and the inverse of an orthogonal matrix have the same defining conditions, they are also orthogonal. </li>
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<li>Since the transpose and the inverse of an orthogonal matrix have the same defining conditions, they are also orthogonal. </li>
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<li>Eigenvectors of an orthogonal matrix can be complex, but all of them have<a>magnitude</a>1. </li>
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<li>Eigenvectors of an orthogonal matrix can be complex, but all of them have<a>magnitude</a>1. </li>
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<li>The identity matrix is orthogonal because \(I^T = I\) and \(I · I = I\).</li>
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<li>The identity matrix is orthogonal because \(I^T = I\) and \(I · I = I\).</li>
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</ol><h2>How to Identify Orthogonal Matrices?</h2>
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</ol><h2>How to Identify Orthogonal Matrices?</h2>
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<p>An orthogonal matrix is a square matrix whose product with its transpose results in the identity matrix. A matrix is also orthogonal if the transpose of the matrix and the inverse of the matrix are equal.</p>
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<p>An orthogonal matrix is a square matrix whose product with its transpose results in the identity matrix. A matrix is also orthogonal if the transpose of the matrix and the inverse of the matrix are equal.</p>
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<p>Let’s take a square matrix A having real elements in the n × n order. AT is the transpose of matrix A. According to the definition, if \(A^T = A^{-1}\), then \(A \cdot A^T = I\).</p>
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<p>Let’s take a square matrix A having real elements in the n × n order. AT is the transpose of matrix A. According to the definition, if \(A^T = A^{-1}\), then \(A \cdot A^T = I\).</p>
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<h2>Determinant of Orthogonal Matrix</h2>
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<h2>Determinant of Orthogonal Matrix</h2>
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<p>The<a>determinant</a>of an orthogonal matrix is 1. To prove so, let us consider an orthogonal matrix A. Then by definition, \(AA^T = I\) </p>
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<p>The<a>determinant</a>of an orthogonal matrix is 1. To prove so, let us consider an orthogonal matrix A. Then by definition, \(AA^T = I\) </p>
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<p>Taking determinants on both sides \(det(AA^T) = det(I)\)</p>
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<p>Taking determinants on both sides \(det(AA^T) = det(I)\)</p>
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<p> The determinant of an identity matrix is 1. </p>
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<p> The determinant of an identity matrix is 1. </p>
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<p> For an orthogonal matrix A, \(det(A) = 1\):</p>
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<p> For an orthogonal matrix A, \(det(A) = 1\):</p>
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<p> Property of determinants, \(det(AB) = det(A) \cdot det(B)\)</p>
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<p> Property of determinants, \(det(AB) = det(A) \cdot det(B)\)</p>
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<p> Since A is orthogonal, we know that \(A^T = A^{-1}\) So, \(AA^T = I ⇒ det(AA^T) = det(I) = 1\)</p>
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<p> Since A is orthogonal, we know that \(A^T = A^{-1}\) So, \(AA^T = I ⇒ det(AA^T) = det(I) = 1\)</p>
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<p> Using the determinant property, \(det(AA^T) = det(A) \cdot det(A^T) \)</p>
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<p> Using the determinant property, \(det(AA^T) = det(A) \cdot det(A^T) \)</p>
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<p>Another property of determinants is \(det(A^T) = det(A)\) Therefore, \(det(A)^2 = 1 \implies det(A) = ±1\)</p>
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<p>Another property of determinants is \(det(A^T) = det(A)\) Therefore, \(det(A)^2 = 1 \implies det(A) = ±1\)</p>
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<p> So, for an orthogonal matrix A, \(det(A)^2 = 1\) and, \(det(A) = 1\).</p>
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<p> So, for an orthogonal matrix A, \(det(A)^2 = 1\) and, \(det(A) = 1\).</p>
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<h2>Inverse of Orthogonal Matrix</h2>
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<h2>Inverse of Orthogonal Matrix</h2>
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<p>As defined, for any orthogonal matrix A, \(A^{-1} = A^T\).</p>
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<p>As defined, for any orthogonal matrix A, \(A^{-1} = A^T\).</p>
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<p>To prove this, we will use the other definition of orthogonal matrix, i.e., \(AA^T = A^TA = I\) ⇒ Let this be (1)</p>
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<p>To prove this, we will use the other definition of orthogonal matrix, i.e., \(AA^T = A^TA = I\) ⇒ Let this be (1)</p>
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<p> Two matrices A and B are said to be each other’s inverses if \(AB = BA = I\) ⇒ Let this be (2)</p>
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<p> Two matrices A and B are said to be each other’s inverses if \(AB = BA = I\) ⇒ Let this be (2)</p>
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<p> From (1) and (2), we get \(B = A^T\). \(B = A^T\) is equal to \(A^{-1} = A^T\) because B is the inverse of A.</p>
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<p> From (1) and (2), we get \(B = A^T\). \(B = A^T\) is equal to \(A^{-1} = A^T\) because B is the inverse of A.</p>
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<p>Hence, proved that the inverse of an orthogonal matrix is equal to its transpose.</p>
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<p>Hence, proved that the inverse of an orthogonal matrix is equal to its transpose.</p>
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<p><strong>Multiplicative Inverse of Orthogonal Matrices</strong></p>
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<p><strong>Multiplicative Inverse of Orthogonal Matrices</strong></p>
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<p>The inverse of an orthogonal matrix is also orthogonal and is equal to the transpose of the original matrix. This shows that orthogonality is maintained during<a>multiplication</a>and inversion.</p>
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<p>The inverse of an orthogonal matrix is also orthogonal and is equal to the transpose of the original matrix. This shows that orthogonality is maintained during<a>multiplication</a>and inversion.</p>
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<h2>Orthogonal Matrix in Linear Algebra</h2>
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<h2>Orthogonal Matrix in Linear Algebra</h2>
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<p>The<a>term</a>“orthogonal” means perpendicular. Two vectors having a<a>dot product</a>of zero are considered orthogonal. In an orthogonal matrix, each row vector and column vector is a unit vector and perpendicular to every other row or column. </p>
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<p>The<a>term</a>“orthogonal” means perpendicular. Two vectors having a<a>dot product</a>of zero are considered orthogonal. In an orthogonal matrix, each row vector and column vector is a unit vector and perpendicular to every other row or column. </p>
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<p>Consider an orthogonal matrix: </p>
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<p>Consider an orthogonal matrix: </p>
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<p>\(A = \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix}\)</p>
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<p>\(A = \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix}\)</p>
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<p> Check for the dot product of the first two rows, it should be zero.</p>
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<p> Check for the dot product of the first two rows, it should be zero.</p>
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<p>Row 1: \({({{1 \over \sqrt 2}, {1\over \sqrt 2}})}\)</p>
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<p>Row 1: \({({{1 \over \sqrt 2}, {1\over \sqrt 2}})}\)</p>
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<p>Row 2: \({({-{1 \over \sqrt 2}, {1\over \sqrt 2}})}\)</p>
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<p>Row 2: \({({-{1 \over \sqrt 2}, {1\over \sqrt 2}})}\)</p>
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<p>Their dot product: \( ({1 \over \sqrt 2} \cdot {-{1 \over \sqrt 2}}) + ({{1 \over \sqrt 2}} \cdot {{1 \over \sqrt 2}}) = -{1\over 2} + {1\over 2}\\ = 0\)</p>
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<p>Their dot product: \( ({1 \over \sqrt 2} \cdot {-{1 \over \sqrt 2}}) + ({{1 \over \sqrt 2}} \cdot {{1 \over \sqrt 2}}) = -{1\over 2} + {1\over 2}\\ = 0\)</p>
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<p>We can see that the first two rows are orthogonal. Keep repeating the process for every two rows and columns. The dot product for each of them should be zero.</p>
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<p>We can see that the first two rows are orthogonal. Keep repeating the process for every two rows and columns. The dot product for each of them should be zero.</p>
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<p>Now, let's find the magnitude of the first row:\(\sqrt{\left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2} = \sqrt{\frac{1}{2} + \frac{1}{2}} = \sqrt{1} = 1\)</p>
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<p>Now, let's find the magnitude of the first row:\(\sqrt{\left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2} = \sqrt{\frac{1}{2} + \frac{1}{2}} = \sqrt{1} = 1\)</p>
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<p>Similarly, the length of every row and column will be 1.</p>
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<p>Similarly, the length of every row and column will be 1.</p>
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<p> </p>
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<p> </p>
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<h2>Tips and Tricks to Master Orthogonal Matrices</h2>
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<h2>Tips and Tricks to Master Orthogonal Matrices</h2>
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<p>An orthogonal matrix is a special square matrix where the transpose is the same as the inverse. It is represented as \(A^TA = AA^T = I\). Here are a few tips and tricks to master orthogonal matrices. </p>
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<p>An orthogonal matrix is a special square matrix where the transpose is the same as the inverse. It is represented as \(A^TA = AA^T = I\). Here are a few tips and tricks to master orthogonal matrices. </p>
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<ul><li>An orthogonal matrix Q satisfies \(Q^TQ=I\), where \(𝑄^T \)is the transpose and I is the identity matrix. </li>
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<ul><li>An orthogonal matrix Q satisfies \(Q^TQ=I\), where \(𝑄^T \)is the transpose and I is the identity matrix. </li>
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<li>To check if a matrix is orthogonal, multiply the given matrix by its transpose. If the result is the identity matrix, it’s orthogonal. </li>
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<li>To check if a matrix is orthogonal, multiply the given matrix by its transpose. If the result is the identity matrix, it’s orthogonal. </li>
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<li>Always remember that the determinant of an orthogonal matrix is always +1 or -1. </li>
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<li>Always remember that the determinant of an orthogonal matrix is always +1 or -1. </li>
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<li>A matrix is orthogonal if the dot product of different rows or columns is 0 (they’re perpendicular) and the dot product of a row or column with itself is 1 (it has unit length). </li>
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<li>A matrix is orthogonal if the dot product of different rows or columns is 0 (they’re perpendicular) and the dot product of a row or column with itself is 1 (it has unit length). </li>
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<li>Start practice with small matrices like \({2 \times 2} {\text { and }} {3 \times 3}\) matrix. Then gradually move to large matrix. </li>
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<li>Start practice with small matrices like \({2 \times 2} {\text { and }} {3 \times 3}\) matrix. Then gradually move to large matrix. </li>
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</ul><h2>Real-Life Applications of Orthogonal Matrix</h2>
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</ul><h2>Real-Life Applications of Orthogonal Matrix</h2>
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<p>Orthogonal matrices are vital in many real-world applications due to their properties of preserving lengths, angles, and orthogonality. Some of these applications are listed below.</p>
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<p>Orthogonal matrices are vital in many real-world applications due to their properties of preserving lengths, angles, and orthogonality. Some of these applications are listed below.</p>
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<ul><li><strong>3D Rotation in Computer Graphics:</strong>Orthogonal matrices are widely used to perform 3D rotations in graphics, animations, and simulations. They preserve shapes, sizes, and angles, ensuring realistic motion without distortion.</li>
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<ul><li><strong>3D Rotation in Computer Graphics:</strong>Orthogonal matrices are widely used to perform 3D rotations in graphics, animations, and simulations. They preserve shapes, sizes, and angles, ensuring realistic motion without distortion.</li>
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</ul><ul><li><strong>Signal Decomposition in Audio and Image Processing: </strong>In orthogonal matrices, each component remains independent, resulting in efficient filtering and compression. It is used in MP3, JPEG, and wireless communication systems.</li>
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</ul><ul><li><strong>Signal Decomposition in Audio and Image Processing: </strong>In orthogonal matrices, each component remains independent, resulting in efficient filtering and compression. It is used in MP3, JPEG, and wireless communication systems.</li>
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</ul><ul><li><strong>Dimensionality Reduction in Machine Learning: </strong>In algorithms like Principal Component Analysis (PCA), principal components are orthogonal vectors. They capture maximum<a>variance</a>without overlaps. This leads to better interpretation and visualization of the<a>data</a>.</li>
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</ul><ul><li><strong>Dimensionality Reduction in Machine Learning: </strong>In algorithms like Principal Component Analysis (PCA), principal components are orthogonal vectors. They capture maximum<a>variance</a>without overlaps. This leads to better interpretation and visualization of the<a>data</a>.</li>
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</ul><ul><li><strong>Attitude Control in Aerospace Engineering: </strong>Orthogonal matrices are used in attitude control systems of satellites, drones, and aircraft to maintain orientation without distortion.</li>
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</ul><ul><li><strong>Attitude Control in Aerospace Engineering: </strong>Orthogonal matrices are used in attitude control systems of satellites, drones, and aircraft to maintain orientation without distortion.</li>
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</ul><ul><li><strong>State Transformations in Quantum Mechanics: </strong>They preserve inner products and probabilities, ensuring physical realism, and hence are used in representing quantum states and transformations in real vector spaces</li>
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</ul><ul><li><strong>State Transformations in Quantum Mechanics: </strong>They preserve inner products and probabilities, ensuring physical realism, and hence are used in representing quantum states and transformations in real vector spaces</li>
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</ul><h2>Common Mistakes and How to Avoid Them in Orthogonal Matrix</h2>
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</ul><h2>Common Mistakes and How to Avoid Them in Orthogonal Matrix</h2>
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<p>Working on problems related to orthogonal matrices might be challenging for some and may lead to mistakes. However, with enough practice, we can overcome challenges and avoid mistakes. In this section, we will look at some of the most common mistakes made by students while working with orthogonal matrix: </p>
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<p>Working on problems related to orthogonal matrices might be challenging for some and may lead to mistakes. However, with enough practice, we can overcome challenges and avoid mistakes. In this section, we will look at some of the most common mistakes made by students while working with orthogonal matrix: </p>
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<h3>Problem 1</h3>
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<h3>Problem 1</h3>
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<p>Verify if this 2x2 matrix is orthogonal</p>
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<p>Verify if this 2x2 matrix is orthogonal</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>Yes, the matrix is orthogonal </p>
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<p>Yes, the matrix is orthogonal </p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p><strong>Given Matrix</strong>: \(A = \begin{bmatrix} cos \theta & - sin \theta \\ sin \theta & cos \theta \\ \end{bmatrix}\)</p>
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<p><strong>Given Matrix</strong>: \(A = \begin{bmatrix} cos \theta & - sin \theta \\ sin \theta & cos \theta \\ \end{bmatrix}\)</p>
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<p>Let θ = 90°</p>
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<p>Let θ = 90°</p>
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<p>\(A = \begin{bmatrix} 0 & - 1 \\ 1 & 0\\ \end{bmatrix}\)</p>
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<p>\(A = \begin{bmatrix} 0 & - 1 \\ 1 & 0\\ \end{bmatrix}\)</p>
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<p><strong>Transpose: </strong></p>
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<p><strong>Transpose: </strong></p>
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<p>\(A ^T= \begin{bmatrix} 0 & 1 \\ -1 & 0\\ \end{bmatrix}\)</p>
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<p>\(A ^T= \begin{bmatrix} 0 & 1 \\ -1 & 0\\ \end{bmatrix}\)</p>
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<p><strong>Product:</strong></p>
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<p><strong>Product:</strong></p>
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<p>\(AA^T = \)\(\begin{bmatrix} 0 & - 1 \\ 1 & 0\\ \end{bmatrix}\) \( \begin{bmatrix} 0 & 1 \\ -1 & 0\\ \end{bmatrix}\)</p>
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<p>\(AA^T = \)\(\begin{bmatrix} 0 & - 1 \\ 1 & 0\\ \end{bmatrix}\) \( \begin{bmatrix} 0 & 1 \\ -1 & 0\\ \end{bmatrix}\)</p>
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<p>⇒ \(A = \begin{bmatrix} 1 & 0 \\ 0 & 1\\ \end{bmatrix}\) = I</p>
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<p>⇒ \(A = \begin{bmatrix} 1 & 0 \\ 0 & 1\\ \end{bmatrix}\) = I</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 2</h3>
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<h3>Problem 2</h3>
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<p>Verify if this 3x3 matrix is orthogonal. A is a rotation matrix around the x-axis</p>
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<p>Verify if this 3x3 matrix is orthogonal. A is a rotation matrix around the x-axis</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>Yes, the matrix is orthogonal. </p>
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<p>Yes, the matrix is orthogonal. </p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p><strong>Given Matrix</strong>: \(A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \\\end{bmatrix}\)</p>
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<p><strong>Given Matrix</strong>: \(A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \\\end{bmatrix}\)</p>
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<p><strong>Transpose: </strong></p>
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<p><strong>Transpose: </strong></p>
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<p>\(A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \\\end{bmatrix}\)</p>
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<p>\(A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \\\end{bmatrix}\)</p>
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<p><strong>Product:</strong></p>
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<p><strong>Product:</strong></p>
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<p>\(AA^T = \)\( \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \\\end{bmatrix}\) \( \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \\\end{bmatrix}\)</p>
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<p>\(AA^T = \)\( \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \\\end{bmatrix}\) \( \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \\\end{bmatrix}\)</p>
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<p>⇒ \(A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{bmatrix}\) = I</p>
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<p>⇒ \(A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{bmatrix}\) = I</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 3</h3>
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<h3>Problem 3</h3>
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<p>Confirm A is orthogonal.</p>
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<p>Confirm A is orthogonal.</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>Yes, A is orthogonal. </p>
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<p>Yes, A is orthogonal. </p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p><strong>Given Matrix</strong>: \(A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \\\end{bmatrix}\)</p>
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<p><strong>Given Matrix</strong>: \(A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \\\end{bmatrix}\)</p>
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<p>Diagonal entries are \(\pm 1\) \( A^T = A AA^T= I A^T = A^{-1} \)</p>
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<p>Diagonal entries are \(\pm 1\) \( A^T = A AA^T= I A^T = A^{-1} \)</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 4</h3>
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<h3>Problem 4</h3>
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<p>Confirm A is orthogonal</p>
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<p>Confirm A is orthogonal</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>Yes, A is orthogonal </p>
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<p>Yes, A is orthogonal </p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p><strong>Given Matrix</strong>: \(A = \begin{bmatrix} \frac{1}{\sqrt 2} & \frac{1}{\sqrt 2} \\[0.3em] \frac{-1}{\sqrt 2} & \frac{1}{\sqrt 2} \\[0.3em] \end{bmatrix}\)</p>
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<p><strong>Given Matrix</strong>: \(A = \begin{bmatrix} \frac{1}{\sqrt 2} & \frac{1}{\sqrt 2} \\[0.3em] \frac{-1}{\sqrt 2} & \frac{1}{\sqrt 2} \\[0.3em] \end{bmatrix}\)</p>
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<p>Check if transpose = Inverse: </p>
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<p>Check if transpose = Inverse: </p>
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<p>\(A^T = \begin{bmatrix} \frac{1}{\sqrt 2} & \frac{-1}{\sqrt 2} \\[0.3em] \frac{1}{\sqrt 2} & \frac{1}{\sqrt 2} \\[0.3em] \end{bmatrix}\)</p>
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<p>\(A^T = \begin{bmatrix} \frac{1}{\sqrt 2} & \frac{-1}{\sqrt 2} \\[0.3em] \frac{1}{\sqrt 2} & \frac{1}{\sqrt 2} \\[0.3em] \end{bmatrix}\)</p>
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<p>\(AA^T = \) \(\begin{bmatrix} 1 & 0 \\[0.3em] 0 & 1 \\[0.3em] \end{bmatrix}\) = I</p>
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<p>\(AA^T = \) \(\begin{bmatrix} 1 & 0 \\[0.3em] 0 & 1 \\[0.3em] \end{bmatrix}\) = I</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 5</h3>
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<h3>Problem 5</h3>
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<p>Verify orthonormal rows</p>
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<p>Verify orthonormal rows</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>All rows are orthonormal </p>
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<p>All rows are orthonormal </p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p><strong>Given Matrix: \(A = \begin{bmatrix} \frac{2}{ 3} & \frac{-2}{3} & \frac{1}{3} \\[0.3em] \frac{1}{3} & \frac{2}{ 3} & \frac{2}{ 3} \\[0.3em] \frac{2}{ 3}& \frac{1}{ 3} & \frac{-2}{ 3}\end{bmatrix}\)</strong></p>
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<p><strong>Given Matrix: \(A = \begin{bmatrix} \frac{2}{ 3} & \frac{-2}{3} & \frac{1}{3} \\[0.3em] \frac{1}{3} & \frac{2}{ 3} & \frac{2}{ 3} \\[0.3em] \frac{2}{ 3}& \frac{1}{ 3} & \frac{-2}{ 3}\end{bmatrix}\)</strong></p>
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<p>The dot product of all the rows is, </p>
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<p>The dot product of all the rows is, </p>
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<p>\({{\big ({ \frac{2}{ 3}} \times {\frac{1}{3}} \big ) + \big({ \frac{-2}{3}} \times {\frac{2}{3}}\big ) + \big({\frac{1}{ 3}} \times {\frac{2}{ 3}} \big ) }}\)</p>
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<p>\({{\big ({ \frac{2}{ 3}} \times {\frac{1}{3}} \big ) + \big({ \frac{-2}{3}} \times {\frac{2}{3}}\big ) + \big({\frac{1}{ 3}} \times {\frac{2}{ 3}} \big ) }}\)</p>
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<p>\(\frac {2} {9} - \frac {4}{9} + \frac{2}{9} = 0\)</p>
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<p>\(\frac {2} {9} - \frac {4}{9} + \frac{2}{9} = 0\)</p>
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<p>Magnitude of row 1:</p>
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<p>Magnitude of row 1:</p>
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<p>\(\sqrt {{({2 \over 3})^2 + ({-2 \over 3})^2 } + ({1 \over 3})^2} = {\sqrt {{4\over {9} } + {4\over {9} } + {1\over {9}} }} = {\sqrt {1 }}=1\)</p>
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<p>\(\sqrt {{({2 \over 3})^2 + ({-2 \over 3})^2 } + ({1 \over 3})^2} = {\sqrt {{4\over {9} } + {4\over {9} } + {1\over {9}} }} = {\sqrt {1 }}=1\)</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h2>FAQs on Orthogonal Matrix</h2>
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<h2>FAQs on Orthogonal Matrix</h2>
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<h3>1.What is the difference between orthogonal matrix and orthonormal</h3>
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<h3>1.What is the difference between orthogonal matrix and orthonormal</h3>
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<p>Orthogonal matrices have perpendicular vectors, i.e., their dot product is zero. Orthonormal means orthogonal, having unit length. </p>
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<p>Orthogonal matrices have perpendicular vectors, i.e., their dot product is zero. Orthonormal means orthogonal, having unit length. </p>
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<h3>2. Types of orthogonal matrix</h3>
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<h3>2. Types of orthogonal matrix</h3>
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<p>Types of orthogonal matrices include: rotation, reflection,<a>permutation</a>, diagonal orthogonal, and block orthogonal matrices. </p>
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<p>Types of orthogonal matrices include: rotation, reflection,<a>permutation</a>, diagonal orthogonal, and block orthogonal matrices. </p>
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<h3>3.How to check an orthogonal matrix?</h3>
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<h3>3.How to check an orthogonal matrix?</h3>
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<p>For a matrix to be orthogonal, either of the following two conditions must be satisfied: </p>
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<p>For a matrix to be orthogonal, either of the following two conditions must be satisfied: </p>
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<ol><li>\(A^T = A^{-1}\), or</li>
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<ol><li>\(A^T = A^{-1}\), or</li>
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<li>\(A^TA = AA^T = I\)</li>
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<li>\(A^TA = AA^T = I\)</li>
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</ol><p>You can also check if all rows and columns are orthonormal. </p>
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</ol><p>You can also check if all rows and columns are orthonormal. </p>
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<h3>4. Is an orthogonal matrix always non-singular?</h3>
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<h3>4. Is an orthogonal matrix always non-singular?</h3>
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<p>Yes, an orthogonal matrix is always non-singular because: </p>
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<p>Yes, an orthogonal matrix is always non-singular because: </p>
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<ul><li>det(A) = ±1, so it is never zero. </li>
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<ul><li>det(A) = ±1, so it is never zero. </li>
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<li>It always has an inverse. </li>
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<li>It always has an inverse. </li>
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</ul><h3>5. Is an orthogonal matrix never symmetric?</h3>
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</ul><h3>5. Is an orthogonal matrix never symmetric?</h3>
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<p> No, some orthogonal matrices can be symmetric, like the identity matrix I. In general, </p>
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<p> No, some orthogonal matrices can be symmetric, like the identity matrix I. In general, </p>
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<ul><li>\(A^T = A → {\text {symmetric}}\)</li>
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<ul><li>\(A^T = A → {\text {symmetric}}\)</li>
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<li>\(A^T = A-1 → {\text {orthogonal}}\)</li>
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<li>\(A^T = A-1 → {\text {orthogonal}}\)</li>
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</ul><p> When both conditions are true, a matrix is both symmetric and orthogonal.</p>
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</ul><p> When both conditions are true, a matrix is both symmetric and orthogonal.</p>
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<h3>6.Why is the Factor Theorem important for students?</h3>
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<h3>6.Why is the Factor Theorem important for students?</h3>
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<p>The<a>factor theorem</a>is important for students as it help students to factor<a>polynomials</a>and to find the n roots of the polynomial. </p>
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<p>The<a>factor theorem</a>is important for students as it help students to factor<a>polynomials</a>and to find the n roots of the polynomial. </p>
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<h3>7.Where is the Factor Theorem used in real life?</h3>
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<h3>7.Where is the Factor Theorem used in real life?</h3>
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<p>Factorial theorem is used in real life in the fields like physics, economics, robotics, and computer graphics. </p>
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<p>Factorial theorem is used in real life in the fields like physics, economics, robotics, and computer graphics. </p>
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<h2>Jaskaran Singh Saluja</h2>
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<h2>Jaskaran Singh Saluja</h2>
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<h3>About the Author</h3>
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<h3>About the Author</h3>
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<p>Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.</p>
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<p>Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.</p>
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<h3>Fun Fact</h3>
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<h3>Fun Fact</h3>
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<p>: He loves to play the quiz with kids through algebra to make kids love it.</p>
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<p>: He loves to play the quiz with kids through algebra to make kids love it.</p>