Chapter 6: Chapter 6

Problem 157

Let $\mathrm{A}=\begin{array}{ccc}\mid 1 & 1 & 1 \mid \\ \mid 0 & 1 & 0 \\\ \mid 0 & 1 & 1\end{array}$ Find the inverse of $\mathrm{A}$.

Short Answer

Expert verified

The inverse of matrix A is: A⁻¹ = \[\begin{array}{ccc}\mid 1 & -1 & 1 \mid \\\ \mid 0 & 1 & 0 \\\ \mid 0 & -1 & 0\end{array}\]

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Step by step solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Calculate the determinant of A

First, we need to find the determinant of matrix A. For a 3x3 matrix, the determinant can be calculated using the diagonal method. \[ det(A) = a_{1,1}(a_{2,2}a_{3,3} - a_{2,3}a_{3,2}) - a_{1,2}(a_{2,1}a_{3,3} - a_{2,3}a_{3,1}) + a_{1,3}(a_{2,1}a_{3,2} - a_{2,2}a_{3,1}) \] Plugging in the values from matrix A: det(A) = $ 1(1\cdot1 - 0\cdot1) - 1(0\cdot1 - 0\cdot0) + 1(0\cdot0 - 1\cdot0) = 1$ Since the determinant is non-zero (det(A) = 1), we can find the inverse of A.

Step 2: Find the matrix of minors

To find the matrix of minors, replace each entry a_{i,j} in matrix A with its corresponding minor, which is the determinant of the 2x2 matrix formed by deleting the row i and column j in the given matrix. Matrix of minors(M) = \[\begin{array}{ccc}\mid 1 & 0 & 0 \mid \\\ \mid 1 & 1 & 1 \\\ \mid 1 & 0 & 0\end{array}\]

Step 3: Find the matrix of cofactors

Next, find the matrix of cofactors by applying a checkerboard pattern of positive and negative signs to the matrix M: Cofactor matrix(C) = \[\begin{array}{ccc}\mid 1 & 0 & 0 \mid \\\ \mid -1 & 1 & -1 \\\ \mid 1 & 0 & 0\end{array}\]

Step 4: Transpose the matrix of cofactors (adjugate matrix)

Transpose the matrix of cofactors to get the adjugate matrix. The adjugate matrix (adj(A)) is the transpose of the matrix of cofactors: adj(A) = \[ \begin{array}{ccc}\mid 1 & -1 & 1 \mid \\\ \mid 0 & 1 & 0 \\\ \mid 0 & -1 & 0\end{array} \]

Step 5: Multiply the adjugate matrix by the reciprocal of the determinant

Finally, to find the inverse of matrix A, multiply the adjugate matrix by the reciprocal of the determinant: A⁻¹ = $\frac{1}{det(A)}$ * adj(A) = $ \frac{1}{1} $ * \[ \begin{array}{ccc}\mid 1 & -1 & 1 \mid \\\ \mid 0 & 1 & 0 \\\ \mid 0 & -1 & 0\end{array} \] A⁻¹ = \[\begin{array}{ccc}\mid 1 & -1 & 1 \mid \\\ \mid 0 & 1 & 0 \\\ \mid 0 & -1 & 0\end{array}\] So, the inverse of matrix A is: A⁻¹ = \[\begin{array}{ccc}\mid 1 & -1 & 1 \mid \\\ \mid 0 & 1 & 0 \\\ \mid 0 & -1 & 0\end{array}\]

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Chapter 6: Chapter 6

Let $\mathrm{A}=\begin{array}{ccc}\mid 1 & 1 & 1 \mid \\ \mid 0 & 1 & 0 \\\ \mid 0 & 1 & 1\end{array}$ Find the inverse of \(\mathrm{A}\).

Short Answer

Step by step solution

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Step 1: Calculate the determinant of A

Step 2: Find the matrix of minors

Step 3: Find the matrix of cofactors

Step 4: Transpose the matrix of cofactors (adjugate matrix)

Step 5: Multiply the adjugate matrix by the reciprocal of the determinant

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Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

Chapter 17

Chapter 18

Chapter 19

Chapter 20

Chapter 21

Chapter 22

Chapter 23

Chapter 24

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