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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>The Cayley-Hamilton theorem, developed by the mathematician Arthur Cayley, is an important concept in matrix algebra. This theorem states that every square matrix satisfies its characteristic equation, which is derived from its characteristic polynomial. This theorem is useful in various mathematical applications, such as finding the inverse of a matrix and computing higher powers of a matrix. We will learn more about the Cayley-Hamilton theorem in this article.</p>
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<p>The Cayley-Hamilton theorem, developed by the mathematician Arthur Cayley, is an important concept in matrix algebra. This theorem states that every square matrix satisfies its characteristic equation, which is derived from its characteristic polynomial. This theorem is useful in various mathematical applications, such as finding the inverse of a matrix and computing higher powers of a matrix. We will learn more about the Cayley-Hamilton theorem in this article.</p>
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<h2>Cayley-Hamilton Theorem</h2>
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<h2>Cayley-Hamilton Theorem</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>The Cayley-Hamilton theorem, developed by the mathematician Arthur Cayley, is an important concept in matrix<a>algebra</a>. This theorem states that every<a>square</a>matrix satisfies its characteristic<a>equation</a>, which is derived from its characteristic<a>polynomial</a>.</p>
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<p>The Cayley-Hamilton theorem, developed by the mathematician Arthur Cayley, is an important concept in matrix<a>algebra</a>. This theorem states that every<a>square</a>matrix satisfies its characteristic<a>equation</a>, which is derived from its characteristic<a>polynomial</a>.</p>
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<p>This theorem is useful in various mathematical applications, such as finding the<a>inverse of a matrix</a>and computing higher<a>powers</a>of a matrix. We will learn more about the Cayley-Hamilton theorem in this article.</p>
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<p>This theorem is useful in various mathematical applications, such as finding the<a>inverse of a matrix</a>and computing higher<a>powers</a>of a matrix. We will learn more about the Cayley-Hamilton theorem in this article.</p>
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<h2>What is the Cayley-Hamilton Theorem?</h2>
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<h2>What is the Cayley-Hamilton Theorem?</h2>
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<p>The Cayley-Hamilton theorem states that every square matrix satisfies its characteristic equation. This equation is derived from the matrix’s characteristic polynomial. To obtain the characteristic polynomial, subtract the scalar times the<a>identity matrix</a>from the given matrix and then computes the<a>determinant</a>of the resulting matrix.</p>
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<p>The Cayley-Hamilton theorem states that every square matrix satisfies its characteristic equation. This equation is derived from the matrix’s characteristic polynomial. To obtain the characteristic polynomial, subtract the scalar times the<a>identity matrix</a>from the given matrix and then computes the<a>determinant</a>of the resulting matrix.</p>
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<p>Eigenvalues are the values of , that make the polynomial equal to zero, and they are special<a>numbers</a>linked to the matrix. In simple words, the Cayley-Hamilton theorem says that every square matrix satisfies an equation that is formed using that very matrix. </p>
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<p>Eigenvalues are the values of , that make the polynomial equal to zero, and they are special<a>numbers</a>linked to the matrix. In simple words, the Cayley-Hamilton theorem says that every square matrix satisfies an equation that is formed using that very matrix. </p>
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<h2>Cayley-Hamilton Theorem Statement</h2>
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<h2>Cayley-Hamilton Theorem Statement</h2>
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<p>The Cayley-Hamilton theorem states that if we have a square matrix made up of real or<a>complex numbers</a>, there is a characteristic polynomial that comes from that matrix. When we plug the matrix into this polynomial, the result will be a zero matrix. The characteristic polynomial of an n × n matrix can be found by using the<a>formula</a>:</p>
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<p>The Cayley-Hamilton theorem states that if we have a square matrix made up of real or<a>complex numbers</a>, there is a characteristic polynomial that comes from that matrix. When we plug the matrix into this polynomial, the result will be a zero matrix. The characteristic polynomial of an n × n matrix can be found by using the<a>formula</a>:</p>
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<p>p() = det(In - A)</p>
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<p>p() = det(In - A)</p>
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<p>Here is a<a>variable</a>.</p>
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<p>Here is a<a>variable</a>.</p>
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<p>It is the identity matrix.</p>
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<p>It is the identity matrix.</p>
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<p>A is the given matrix,</p>
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<p>A is the given matrix,</p>
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<p>det means the determinant of that<a>expression</a>.</p>
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<p>det means the determinant of that<a>expression</a>.</p>
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<p>The characteristic polynomial looks like: p() = n + an - 1n - 1 + … + a1 + a0</p>
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<p>The characteristic polynomial looks like: p() = n + an - 1n - 1 + … + a1 + a0</p>
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<p>Here, the highest power of is n, and its<a>coefficient</a>is always 1. The other<a>terms</a>have their<a>constants</a>.</p>
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<p>Here, the highest power of is n, and its<a>coefficient</a>is always 1. The other<a>terms</a>have their<a>constants</a>.</p>
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<p>According to the theorem, if we replace with the matrix A, the equation becomes:</p>
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<p>According to the theorem, if we replace with the matrix A, the equation becomes:</p>
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<p>p(A) = An + an - 1An - 1 + … + a1A + a0In, we will get 0 always.</p>
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<p>p(A) = An + an - 1An - 1 + … + a1A + a0In, we will get 0 always.</p>
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<h2>Cayley-Hamilton Theorem Formula</h2>
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<h2>Cayley-Hamilton Theorem Formula</h2>
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<p>The Cayley-Hamilton theorem helps simplify complex calculations and can also be used to find the inverse of a matrix efficiently. The formula for the Cayley-Hamilton theorem states that, for any n × n matrix, its characteristic polynomial looks like:</p>
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<p>The Cayley-Hamilton theorem helps simplify complex calculations and can also be used to find the inverse of a matrix efficiently. The formula for the Cayley-Hamilton theorem states that, for any n × n matrix, its characteristic polynomial looks like:</p>
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<p>p() = n + an - 1n - 1 + … + a1 + a0</p>
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<p>p() = n + an - 1n - 1 + … + a1 + a0</p>
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<p>The Cayley-Hamilton theorem says: p(A) = An + an - 1An - 1 + … + a1A + a0I</p>
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<p>The Cayley-Hamilton theorem says: p(A) = An + an - 1An - 1 + … + a1A + a0I</p>
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<p>Here, A represents the given square matrix. I is the identity matrix. </p>
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<p>Here, A represents the given square matrix. I is the identity matrix. </p>
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<h3>Cayley-Hamilton Theorem 2 × 2</h3>
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<h3>Cayley-Hamilton Theorem 2 × 2</h3>
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<p>To apply the Cayley-Hamilton theorem to a 2 × 2 matrix, follow the steps given below:</p>
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<p>To apply the Cayley-Hamilton theorem to a 2 × 2 matrix, follow the steps given below:</p>
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<p><strong>Step 1:</strong>Take a 2 × 2 matrix.</p>
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<p><strong>Step 1:</strong>Take a 2 × 2 matrix.</p>
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<p><strong>Step 2:</strong>Find its characteristic equation.</p>
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<p><strong>Step 2:</strong>Find its characteristic equation.</p>
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<p><strong>Step 3:</strong>For a 2 × 2 matrix, the characteristic equation is:</p>
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<p><strong>Step 3:</strong>For a 2 × 2 matrix, the characteristic equation is:</p>
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<p>2 - S1 + S0 = 0</p>
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<p>2 - S1 + S0 = 0</p>
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<p>Here, S1 =<a>sum</a>of the diagonal numbers (trace of the matrix),</p>
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<p>Here, S1 =<a>sum</a>of the diagonal numbers (trace of the matrix),</p>
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<p>S2 = determinant of the matrix.</p>
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<p>S2 = determinant of the matrix.</p>
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<p><strong>Step 4:</strong>According to the Cayley-Hamilton theorem, replace with the matrix B. The equation becomes:</p>
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<p><strong>Step 4:</strong>According to the Cayley-Hamilton theorem, replace with the matrix B. The equation becomes:</p>
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<p>B2 - S1B + S0I = 0</p>
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<p>B2 - S1B + S0I = 0</p>
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<p>Where I is the identity matrix of the same order.</p>
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<p>Where I is the identity matrix of the same order.</p>
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<p>This means if we plug the matrix into its equation, we get a zero matrix.</p>
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<p>This means if we plug the matrix into its equation, we get a zero matrix.</p>
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<h2>Cayley-Hamilton Theorem 3 × 3</h2>
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<h2>Cayley-Hamilton Theorem 3 × 3</h2>
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<p>For a 3 × 3 matrix, first find the characteristic polynomial, which looks like: p() = 3 - T22 + T1 - T0 = 0</p>
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<p>For a 3 × 3 matrix, first find the characteristic polynomial, which looks like: p() = 3 - T22 + T1 - T0 = 0</p>
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<p>Where, T2 = sum of the diagonal numbers,</p>
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<p>Where, T2 = sum of the diagonal numbers,</p>
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<p>T1 = sum of all 2 × 2<a>minors</a>taken from the main diagonal elements,</p>
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<p>T1 = sum of all 2 × 2<a>minors</a>taken from the main diagonal elements,</p>
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<p>T0 = determinant of the matrix.</p>
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<p>T0 = determinant of the matrix.</p>
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<p>If we put the 3 × 3 matrix into its characteristic equation, the answer is always zero. Apply the Cayley-Hamilton theorem, by replacing with the matrix C, the equation becomes,</p>
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<p>If we put the 3 × 3 matrix into its characteristic equation, the answer is always zero. Apply the Cayley-Hamilton theorem, by replacing with the matrix C, the equation becomes,</p>
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<p>C3 - T2C2 + T1C - T0I = 0</p>
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<p>C3 - T2C2 + T1C - T0I = 0</p>
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<p>Where I is the identity matrix of order 3.</p>
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<p>Where I is the identity matrix of order 3.</p>
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<p>This means that every 3 × 3 matrix satisfies its characteristic equation.</p>
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<p>This means that every 3 × 3 matrix satisfies its characteristic equation.</p>
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<h2>Cayley-Hamilton Theorem Proof</h2>
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<h2>Cayley-Hamilton Theorem Proof</h2>
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<p>There are several methods to prove the Cayley-Hamilton theorem, but the easiest method is by using substitution. Let the matrix be, A = ca db The theorem states that: p(A) = A2 - (a + d)A + (ad - bc)I = 0</p>
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<p>There are several methods to prove the Cayley-Hamilton theorem, but the easiest method is by using substitution. Let the matrix be, A = ca db The theorem states that: p(A) = A2 - (a + d)A + (ad - bc)I = 0</p>
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<p>Step 1: Find A2 </p>
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<p>Step 1: Find A2 </p>
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<p>Step 2: Find (a + d)A</p>
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<p>Step 2: Find (a + d)A</p>
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<p>Step 3: Find (ad - bc)I</p>
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<p>Step 3: Find (ad - bc)I</p>
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<p>Step 4: Combine them Hence, the theorem is proved.</p>
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<p>Step 4: Combine them Hence, the theorem is proved.</p>
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<h2>Common Mistakes and How To Avoid Them in Cayley-Hamilton Theorem</h2>
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<h2>Common Mistakes and How To Avoid Them in Cayley-Hamilton Theorem</h2>
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<p>The Cayley-Hamilton theorem is an important concept in linear algebra. While applying this theorem, students often make mistakes due to misunderstanding the steps or missing key details. Given below are some of the common mistakes and the ways to avoid them.</p>
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<p>The Cayley-Hamilton theorem is an important concept in linear algebra. While applying this theorem, students often make mistakes due to misunderstanding the steps or missing key details. Given below are some of the common mistakes and the ways to avoid them.</p>
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<h2>Real Life Applications of Cayley-Hamilton Theorem</h2>
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<h2>Real Life Applications of Cayley-Hamilton Theorem</h2>
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<p>The Cayley-Hamilton theorem is used in areas like engineering, physics, computer graphics, and economics to solve matrix problems quickly. It also helps to make big calculations easier, like finding inverses or powers. Here are some real-life examples of the Cayley-Hamilton theorem.</p>
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<p>The Cayley-Hamilton theorem is used in areas like engineering, physics, computer graphics, and economics to solve matrix problems quickly. It also helps to make big calculations easier, like finding inverses or powers. Here are some real-life examples of the Cayley-Hamilton theorem.</p>
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<ul><li><strong>Control Systems:</strong>In control systems, the Cayley-Hamilton theorem simplifies matrix calculations for analyzing and designing systems, such as robots, cruise control, and air conditioning, by making it easier to compute matrix powers for predicting system behavior. </li>
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<ul><li><strong>Control Systems:</strong>In control systems, the Cayley-Hamilton theorem simplifies matrix calculations for analyzing and designing systems, such as robots, cruise control, and air conditioning, by making it easier to compute matrix powers for predicting system behavior. </li>
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</ul><ul><li><strong>Electrical Circuits:</strong>In analyzing circuits with<a>multiple</a>loops, this theorem helps to solve differential equations that describe how voltage and current change over time using the matrix exponential. </li>
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</ul><ul><li><strong>Electrical Circuits:</strong>In analyzing circuits with<a>multiple</a>loops, this theorem helps to solve differential equations that describe how voltage and current change over time using the matrix exponential. </li>
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</ul><ul><li><strong>Satellite Systems:</strong>In satellite trajectory prediction, the Cayley-Hamilton theorem simplifies the complex matrices used in motion modeling, making it easier to calculate accurate orbits.</li>
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</ul><ul><li><strong>Satellite Systems:</strong>In satellite trajectory prediction, the Cayley-Hamilton theorem simplifies the complex matrices used in motion modeling, making it easier to calculate accurate orbits.</li>
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</ul><h3>Problem 1</h3>
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</ul><h3>Problem 1</h3>
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<p>Verify the Cayley-Hamilton theorem for A = 31 42</p>
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<p>Verify the Cayley-Hamilton theorem for A = 31 42</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p> A2 - 5A - 2I = 00 00</p>
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<p> A2 - 5A - 2I = 00 00</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Trace = 1 + 4 = 5</p>
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<p>Trace = 1 + 4 = 5</p>
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<p>Determinant = (1)(4) - (2)(3) = -3</p>
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<p>Determinant = (1)(4) - (2)(3) = -3</p>
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<p>Characteristic equation: 2 - 5 - 2 = 0</p>
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<p>Characteristic equation: 2 - 5 - 2 = 0</p>
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<p>Theorem: A2 - 5A - 2I = 0</p>
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<p>Theorem: A2 - 5A - 2I = 0</p>
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<p>A2 = 31 42 × 31 42 = 157 2210</p>
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<p>A2 = 31 42 × 31 42 = 157 2210</p>
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<p>-5A = -15-5 -20-10</p>
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<p>-5A = -15-5 -20-10</p>
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<p>-2I = 0-2 -20</p>
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<p>-2I = 0-2 -20</p>
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<p>Add:</p>
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<p>Add:</p>
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<p>157 2210 + -15-5 -20-10 + 0-2 -20 = 00 00</p>
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<p>157 2210 + -15-5 -20-10 + 0-2 -20 = 00 00</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 the Cayley Hamilton theorem for B = 02 31</p>
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<p>Verify the Cayley Hamilton theorem for B = 02 31</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>B2 - 5B + 6I = 00 00</p>
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<p>B2 - 5B + 6I = 00 00</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Trace of the given matrix is 2 + 3 = 5</p>
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<p>Trace of the given matrix is 2 + 3 = 5</p>
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<p>Determinant of the given matrix is 2 × 3 - 0 × 1 = 6</p>
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<p>Determinant of the given matrix is 2 × 3 - 0 × 1 = 6</p>
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<p>Therefore, the characteristic equation is 2 - 5 + 6 = 0</p>
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<p>Therefore, the characteristic equation is 2 - 5 + 6 = 0</p>
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<p>Apply the theorem, B2 - 5B + 6I </p>
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<p>Apply the theorem, B2 - 5B + 6I </p>
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<p>Calculate the theorem,</p>
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<p>Calculate the theorem,</p>
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<p>Add:</p>
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<p>Add:</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>Verify the Cayley-Hamilton theorem for</p>
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<p>Verify the Cayley-Hamilton theorem for</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Trace = 1 + 2 = 3</p>
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<p>Trace = 1 + 2 = 3</p>
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<p>Determinant = 1 × 2 - 0 × 1 = 2</p>
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<p>Determinant = 1 × 2 - 0 × 1 = 2</p>
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<p>Equation = 2 - 3 + 2 = 0</p>
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<p>Equation = 2 - 3 + 2 = 0</p>
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<p>Apply the theorem, C2 - 3C + 2I </p>
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<p>Apply the theorem, C2 - 3C + 2I </p>
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<p>Calculate the values for C2, - 3C, and 2I ,</p>
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<p>Calculate the values for C2, - 3C, and 2I ,</p>
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<p>Add:</p>
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<p>Add:</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>Verify the Cayley-Hamilton theorem for</p>
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<p>Verify the Cayley-Hamilton theorem for</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Trace = 3 + 1 = 4</p>
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<p>Trace = 3 + 1 = 4</p>
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<p>Determinant = 3 × 1 - 0 × 2 = 3</p>
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<p>Determinant = 3 × 1 - 0 × 2 = 3</p>
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<p>Equation: 2 - 4 + 3 = 0</p>
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<p>Equation: 2 - 4 + 3 = 0</p>
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<p>Applying the theorem: D2 - 4D + 3I</p>
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<p>Applying the theorem: D2 - 4D + 3I</p>
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<p>Calculate:</p>
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<p>Calculate:</p>
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<p> Add all of them,</p>
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<p> Add all of them,</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 the Cayley-Hamilton theorem for</p>
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<p>Verify the Cayley-Hamilton theorem for</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p> The trace is 2 + 2 = 4</p>
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<p> The trace is 2 + 2 = 4</p>
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<p>The determinant is 2 × 2 - 1 × 0 = 4</p>
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<p>The determinant is 2 × 2 - 1 × 0 = 4</p>
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<p>Therefore, the equation becomes 2 - 4 + 4 = 0</p>
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<p>Therefore, the equation becomes 2 - 4 + 4 = 0</p>
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<p>Apply theorem: E2 - 4E + 4I</p>
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<p>Apply theorem: E2 - 4E + 4I</p>
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<p>Calculate:</p>
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<p>Calculate:</p>
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<p>Add,</p>
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<p>Add,</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h2>FAQs on Cayley-Hamilton Theorem</h2>
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<h2>FAQs on Cayley-Hamilton Theorem</h2>
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<h3>1.What is the Cayley-Hamilton theorem?</h3>
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<h3>1.What is the Cayley-Hamilton theorem?</h3>
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<p>Cayley-Hamilton theorem in<a>linear algebra</a>states that every square matrix satisfies its characteristic equation. </p>
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<p>Cayley-Hamilton theorem in<a>linear algebra</a>states that every square matrix satisfies its characteristic equation. </p>
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<h3>2.Why is the Cayley-Hamilton theorem useful?</h3>
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<h3>2.Why is the Cayley-Hamilton theorem useful?</h3>
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<p>It is useful for calculating higher powers of a matrix, finding the inverse of a matrix, and simplifying the process of solving matrix equations. </p>
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<p>It is useful for calculating higher powers of a matrix, finding the inverse of a matrix, and simplifying the process of solving matrix equations. </p>
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<h3>3.Can the Cayley-Hamilton theorem be used for any matrices?</h3>
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<h3>3.Can the Cayley-Hamilton theorem be used for any matrices?</h3>
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<p>No, the Cayley-Hamilton theorem works only for square matrices.</p>
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<p>No, the Cayley-Hamilton theorem works only for square matrices.</p>
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<h3>4.Who discovered the Cayley-Hamilton Theorem?</h3>
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<h3>4.Who discovered the Cayley-Hamilton Theorem?</h3>
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<p>The Cayley-Hamilton theorem was proposed by the mathematician Arthur Cayley and later formalized by William Rowan Hamilton. </p>
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<p>The Cayley-Hamilton theorem was proposed by the mathematician Arthur Cayley and later formalized by William Rowan Hamilton. </p>
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<h3>5.. Is this theorem only for real numbers?</h3>
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<h3>5.. Is this theorem only for real numbers?</h3>
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<p>No, it works for matrices with real or complex numbers.</p>
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<p>No, it works for matrices with real or complex numbers.</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>