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( -10! BIf (21-A)-! evaluate the matrix A.

( -10! BIf (21-A)-! evaluate the matrix A.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem Statement:** Given the equation \((21 - A)^{-1} = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 1 \\ -1 & 1 & 0 \end{bmatrix}\), evaluate the matrix \(A\). **Solution Steps:** 1. **Understanding Inverse Matrices:** - The matrix \((21 - A)^{-1}\) is the inverse of the matrix \(21 - A\). - This implies that \(21 - A\) is equal to the inverse of the given matrix. 2. **Compute the Matrix \(21 - A\):** - Find the inverse of the matrix \(\begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 1 \\ -1 & 1 & 0 \end{bmatrix}\). 3. **Calculate the Inverse:** - Use standard methods for finding the inverse of a 3x3 matrix (e.g., using determinants and minors or row reduction). 4. **Subtract and Solve for \(A\):** - After computing \((21 - A)\), solve the equation to find \(A\) by rearranging \(A = 21I - (21 - A)\), where \(I\) is the identity matrix. 5. **Verification:** - Ensure that your calculated \(A\) satisfies the original equation by checking if the inverse operation returns the given matrix. **Conclusion:** The steps outlined provide a systematic method to find matrix \(A\) that will satisfy the given condition within the problem statement.
Transcribed Image Text:**Problem Statement:** Given the equation \((21 - A)^{-1} = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 1 \\ -1 & 1 & 0 \end{bmatrix}\), evaluate the matrix \(A\). **Solution Steps:** 1. **Understanding Inverse Matrices:** - The matrix \((21 - A)^{-1}\) is the inverse of the matrix \(21 - A\). - This implies that \(21 - A\) is equal to the inverse of the given matrix. 2. **Compute the Matrix \(21 - A\):** - Find the inverse of the matrix \(\begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 1 \\ -1 & 1 & 0 \end{bmatrix}\). 3. **Calculate the Inverse:** - Use standard methods for finding the inverse of a 3x3 matrix (e.g., using determinants and minors or row reduction). 4. **Subtract and Solve for \(A\):** - After computing \((21 - A)\), solve the equation to find \(A\) by rearranging \(A = 21I - (21 - A)\), where \(I\) is the identity matrix. 5. **Verification:** - Ensure that your calculated \(A\) satisfies the original equation by checking if the inverse operation returns the given matrix. **Conclusion:** The steps outlined provide a systematic method to find matrix \(A\) that will satisfy the given condition within the problem statement.
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