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1 - <p>141 Learners</p>

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2 <p>Last updated on<strong>November 25, 2025</strong></p>

3 <p>The identity matrix is unique because it acts like the number 1 in matrix multiplication. Its inverse is the identity matrix itself, since multiplying it by itself gives the same matrix. In this lesson, we’ll explore why this property holds true.</p>

4 <h2>What is the Inverse of Identity Matrix?</h2>

5 <p>What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math</p>

6 <p>▶</p>

7 <p>An<a>identity matrix</a>is a<a>square</a>matrix that has 1s on the main diagonal,<a>i</a>.e., from top-left to bottom-right, and 0s in all other places.</p>

8 <p>The inverse<a>of</a>the identity matrix is the identity matrix itself. This is because multiplying any identity matrix by itself results in the identity matrix, just like 1 × 1 = 1 in<a>arithmetic</a>.</p>

9 <p>Hence, the inverse of the identity matrix is the identity matrix itself. For example, an identity matrix of order 2 is:</p>

10 <p>I2 = 01 10</p>

11 <h2>Inverse of Identity Matrix Of Order n</h2>

12 <p>A<a>formula</a>for the inverse of any square matrix A is:</p>

13 <p>A-1 = 1A × adj(A)</p>

14 <p>For an identity matrix, In:</p>

15 <p>The<a>determinant</a>is always 1, |In| = 1.</p>

16 <p>Its adjoint is itself, adj(In) = In. </p>

17 <p>In-1 = 11In = In</p>

18 <p>Therefore, the inverse of the identity matrix of order n is the same identity matrix.</p>

19 <h2>Inverse of Identity Matrix Of Order 2</h2>

20 <p>The identity matrix of order 2 is:</p>

21 <p>I2 = 01 10</p>

22 <p>Determinant: |I2| = 1</p>

23 <p>Adjoint: adj(I2) = 01 10</p>

24 <p>The inverse of the identity matrix of order 2:</p>

25 <p>I2-1 = 11I2 = I2</p>

26 <p>01 10 × 01 10 = 01 10</p>

27 <p>So, the inverse of a 2 × 2 identity matrix is the same 2 × 2 identity matrix. </p>

28 <h3>Explore Our Programs</h3>

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30 <h2>Inverse of Identity Matrix Of Order 3</h2>

29 <h2>Inverse of Identity Matrix Of Order 3</h2>

31 <p>The matrix is I3 = </p>

30 <p>The matrix is I3 = </p>

32 <p>1</p>

31 <p>1</p>

33 <p>0</p>

32 <p>0</p>

34 <p>0</p>

33 <p>0</p>

35 <p>0</p>

34 <p>0</p>

36 <p>1</p>

35 <p>1</p>

37 <p>0</p>

36 <p>0</p>

38 <p>0</p>

37 <p>0</p>

39 <p>0</p>

38 <p>0</p>

40 <p>1</p>

39 <p>1</p>

41 <p>Determinant: |I3| = 1</p>

40 <p>Determinant: |I3| = 1</p>

42 <p>Adjoint: adj(I3) = I3</p>

41 <p>Adjoint: adj(I3) = I3</p>

43 <p>The inverse of the matrix is:</p>

42 <p>The inverse of the matrix is:</p>

44 <p>I3-1 = 11I3 = I3</p>

43 <p>I3-1 = 11I3 = I3</p>

45 <p>Multiplying I3 by itself gives I3. Therefore, the inverse of the identity matrix of order 3 is equal to the identity matrix of order 3.</p>

44 <p>Multiplying I3 by itself gives I3. Therefore, the inverse of the identity matrix of order 3 is equal to the identity matrix of order 3.</p>

46 <h2>Common Mistakes and How to Avoid Them in Inverse of Identity Matrix</h2>

45 <h2>Common Mistakes and How to Avoid Them in Inverse of Identity Matrix</h2>

47 <p>Students often make mistakes when dealing with the inverse of the identity matrix by confusing it with general matrix inversion. These are some of the common mistakes that we can avoid in the future. </p>

46 <p>Students often make mistakes when dealing with the inverse of the identity matrix by confusing it with general matrix inversion. These are some of the common mistakes that we can avoid in the future. </p>

48 <h2>Real Life Applications of Inverse of Identity Matrix</h2>

47 <h2>Real Life Applications of Inverse of Identity Matrix</h2>

49 <p>The identity matrix is a special type of square matrix with 1s on the main diagonal and 0s in all other positions. One of the unique properties of the identity matrix is that its inverse is itself. Here are some of the real-life applications of the inverse of the identity matrix.</p>

48 <p>The identity matrix is a special type of square matrix with 1s on the main diagonal and 0s in all other positions. One of the unique properties of the identity matrix is that its inverse is itself. Here are some of the real-life applications of the inverse of the identity matrix.</p>

50 <ul><li><strong>Computer Graphics:</strong>In 3D graphics, identity matrices are used as the starting points for transformations such as rotation, scaling, or translation. The inverse of the identity matrix is used to reset the transformations and return the objects to their original state. For example, when designing a video game, multiplying a 3D object by In-1 = In, leaves it unchanged. </li>

49 <ul><li><strong>Computer Graphics:</strong>In 3D graphics, identity matrices are used as the starting points for transformations such as rotation, scaling, or translation. The inverse of the identity matrix is used to reset the transformations and return the objects to their original state. For example, when designing a video game, multiplying a 3D object by In-1 = In, leaves it unchanged. </li>

51 </ul><ul><li><strong>Robotics and Control Systems:</strong>Robots use identity matrices to represent the default position or no movement. The inverse of the identity matrix is used to reset the robot’s position during calculations of joint movements.</li>

50 </ul><ul><li><strong>Robotics and Control Systems:</strong>Robots use identity matrices to represent the default position or no movement. The inverse of the identity matrix is used to reset the robot’s position during calculations of joint movements.</li>

52 </ul><ul><li><strong>Programming and Algorithms:</strong>In programming tools like NumPy or MATLAB, identity matrices are often used as default inputs in matrix operations. Since the identity matrix doesn’t change other matrices when multiplied, it simplifies algorithm design and debugging.</li>

51 </ul><ul><li><strong>Programming and Algorithms:</strong>In programming tools like NumPy or MATLAB, identity matrices are often used as default inputs in matrix operations. Since the identity matrix doesn’t change other matrices when multiplied, it simplifies algorithm design and debugging.</li>

53 </ul><h3>Problem 1</h3>

52 </ul><h3>Problem 1</h3>

54 <p>Find the inverse of the identity matrix of order 3.</p>

53 <p>Find the inverse of the identity matrix of order 3.</p>

55 <p>Okay, lets begin</p>

54 <p>Okay, lets begin</p>

56 <p>I3-1 = </p>

55 <p>I3-1 = </p>

57 <p>1</p>

56 <p>1</p>

58 <p>0</p>

57 <p>0</p>

59 <p>0</p>

58 <p>0</p>

60 <p>0</p>

59 <p>0</p>

61 <p>1</p>

60 <p>1</p>

62 <p>0</p>

61 <p>0</p>

63 <p>0</p>

62 <p>0</p>

64 <p>0</p>

63 <p>0</p>

65 <p>1</p>

64 <p>1</p>

66 <h3>Explanation</h3>

65 <h3>Explanation</h3>

67 <p>The identity matrix of order 3 is I3 =</p>

66 <p>The identity matrix of order 3 is I3 =</p>

68 <p>1</p>

67 <p>1</p>

69 <p>0</p>

68 <p>0</p>

70 <p>0</p>

69 <p>0</p>

71 <p>0</p>

70 <p>0</p>

72 <p>1</p>

71 <p>1</p>

73 <p>0</p>

72 <p>0</p>

74 <p>0</p>

73 <p>0</p>

75 <p>0</p>

74 <p>0</p>

76 <p>1</p>

75 <p>1</p>

77 <p>The determinant is |I3| = 1.</p>

76 <p>The determinant is |I3| = 1.</p>

78 <p>The adjoint is adj(I3) = I3.</p>

77 <p>The adjoint is adj(I3) = I3.</p>

79 <p>Using the formula:</p>

78 <p>Using the formula:</p>

80 <p>I3-1 = 1I3 × adj(I3), we get</p>

79 <p>I3-1 = 1I3 × adj(I3), we get</p>

81 <p>I3-1 = 11 × I3 = I3</p>

80 <p>I3-1 = 11 × I3 = I3</p>

82 <p>Well explained 👍</p>

81 <p>Well explained 👍</p>

83 <h3>Problem 2</h3>

82 <h3>Problem 2</h3>

84 <p>Verify that the inverse of the identity matrix of order 2 is the same matrix.</p>

83 <p>Verify that the inverse of the identity matrix of order 2 is the same matrix.</p>

85 <p>Okay, lets begin</p>

84 <p>Okay, lets begin</p>

86 <p>I2-1 = I2</p>

85 <p>I2-1 = I2</p>

87 <h3>Explanation</h3>

86 <h3>Explanation</h3>

88 <p>We have, I2 = 01 10</p>

87 <p>We have, I2 = 01 10</p>

89 <p>Multiply I2 by itself:</p>

88 <p>Multiply I2 by itself:</p>

90 <p>I2 × I2 = 01 10 × 01 10 = 01 10</p>

89 <p>I2 × I2 = 01 10 × 01 10 = 01 10</p>

91 <p>Well explained 👍</p>

90 <p>Well explained 👍</p>

92 <h3>Problem 3</h3>

91 <h3>Problem 3</h3>

93 <p>Is the inverse of I2 = 01 10 is 10 01</p>

92 <p>Is the inverse of I2 = 01 10 is 10 01</p>

94 <p>Okay, lets begin</p>

93 <p>Okay, lets begin</p>

95 <p>No, the inverse of I2 is the same I2 itself.</p>

94 <p>No, the inverse of I2 is the same I2 itself.</p>

96 <h3>Explanation</h3>

95 <h3>Explanation</h3>

97 <p>The inverse of the identity matrix I2 is:</p>

96 <p>The inverse of the identity matrix I2 is:</p>

98 <p>I2-1 = 01 10</p>

97 <p>I2-1 = 01 10</p>

99 <p>The given matrix is incorrect because multiplying I2 by 10 01 does not return I2.</p>

98 <p>The given matrix is incorrect because multiplying I2 by 10 01 does not return I2.</p>

100 <p>Well explained 👍</p>

99 <p>Well explained 👍</p>

101 <h3>Problem 4</h3>

100 <h3>Problem 4</h3>

102 <p>If the inverse of a matrix A is A-1, what is the inverse of the identity matrix In?</p>

101 <p>If the inverse of a matrix A is A-1, what is the inverse of the identity matrix In?</p>

103 <p>Okay, lets begin</p>

102 <p>Okay, lets begin</p>

104 <p><strong> </strong>The inverse of In is In itself.</p>

103 <p><strong> </strong>The inverse of In is In itself.</p>

105 <h3>Explanation</h3>

104 <h3>Explanation</h3>

106 <p>The identity matrix acts like the number 1 in multiplication. Since In × In = In, it is its own inverse.</p>

105 <p>The identity matrix acts like the number 1 in multiplication. Since In × In = In, it is its own inverse.</p>

107 <p>Well explained 👍</p>

106 <p>Well explained 👍</p>

108 <h3>Problem 5</h3>

107 <h3>Problem 5</h3>

109 <p>Find the inverse of the identity matrix of order 4</p>

108 <p>Find the inverse of the identity matrix of order 4</p>

110 <p>Okay, lets begin</p>

109 <p>Okay, lets begin</p>

111 <p>The inverse of I4 is I4 </p>

110 <p>The inverse of I4 is I4 </p>

112 <h3>Explanation</h3>

111 <h3>Explanation</h3>

113 <p>The identity matrix of order n is:</p>

112 <p>The identity matrix of order n is:</p>

114 <p>The determinant of |I4| is 1</p>

113 <p>The determinant of |I4| is 1</p>

115 <p>The adjoint of I4 is I4</p>

114 <p>The adjoint of I4 is I4</p>

116 <p>So, by the inverse formula:</p>

115 <p>So, by the inverse formula:</p>

117 <p>I4-1 = (1 / det) × adjoint = 11 × I4 = I4</p>

116 <p>I4-1 = (1 / det) × adjoint = 11 × I4 = I4</p>

118 <p>Well explained 👍</p>

117 <p>Well explained 👍</p>

119 <h2>FAQs on Inverse of Identity Matrix</h2>

118 <h2>FAQs on Inverse of Identity Matrix</h2>

120 <h3>1.What is the inverse of an identity matrix?</h3>

119 <h3>1.What is the inverse of an identity matrix?</h3>

121 <p>The inverse of an identity matrix is the same identity matrix itself. For example, I2 = 01 10 and I2-1 = 01 10.</p>

120 <p>The inverse of an identity matrix is the same identity matrix itself. For example, I2 = 01 10 and I2-1 = 01 10.</p>

122 <h3>2. Does every identity matrix have an inverse?</h3>

121 <h3>2. Does every identity matrix have an inverse?</h3>

123 <p>Yes, every identity matrix of any order has an inverse, and it is always the same identity matrix. </p>

122 <p>Yes, every identity matrix of any order has an inverse, and it is always the same identity matrix. </p>

124 <h3>3.What is the determinant of an identity matrix?</h3>

123 <h3>3.What is the determinant of an identity matrix?</h3>

125 <p>The determinant of any identity matrix In is always 1. </p>

124 <p>The determinant of any identity matrix In is always 1. </p>

126 <h3>4. Can a zero matrix have an inverse like the identity matrix?</h3>

125 <h3>4. Can a zero matrix have an inverse like the identity matrix?</h3>

127 <p>No, the zero matrix does not have an inverse. Only the identity matrix and other invertible matrices have inverses.</p>

126 <p>No, the zero matrix does not have an inverse. Only the identity matrix and other invertible matrices have inverses.</p>

128 <h3>5.Can non-square matrices have an identity matrix or an inverse?</h3>

127 <h3>5.Can non-square matrices have an identity matrix or an inverse?</h3>

129 <p>No, identity matrices and their inverses only exist for square matrices. </p>

128 <p>No, identity matrices and their inverses only exist for square matrices. </p>

130 <h2>Jaskaran Singh Saluja</h2>

129 <h2>Jaskaran Singh Saluja</h2>

131 <h3>About the Author</h3>

130 <h3>About the Author</h3>

132 <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>

131 <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>

133 <h3>Fun Fact</h3>

132 <h3>Fun Fact</h3>

134 <p>: He loves to play the quiz with kids through algebra to make kids love it.</p>

133 <p>: He loves to play the quiz with kids through algebra to make kids love it.</p>