Question:
If \(A=\begin{bmatrix} 2 & -3 \\ 5 & -7 \end{bmatrix}\), then \(A-A^{-1}=\)
If \(A=\begin{bmatrix} 2 & -3 \\ 5 & -7 \end{bmatrix}\), then \(A-A^{-1}=\)
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Find the inverse using the adjoint formula and subtract it from A.
Updated On: Nov 12, 2026
- \(\begin{bmatrix} -9 & 6 \\ -10 & 9 \end{bmatrix}\)
- \(\begin{bmatrix} 9 & -6 \\ 10 & -9 \end{bmatrix}\)
- \(\begin{bmatrix} -9 & 10 \\ -6 & 9 \end{bmatrix}\)
- \(\begin{bmatrix} 9 & -10 \\ 6 & -9 \end{bmatrix}\)
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept
For a 2 by 2 matrix \(\begin{bmatrix}a&b\\c&d\end{bmatrix}\), the inverse is \(\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}\).
Step 2: Find the determinant
\(|A|=2(-7)-(-3)(5)=-14+15=1\).
Step 3: Find the inverse
\(A^{-1}=\begin{bmatrix}-7&3\\-5&2\end{bmatrix}\).
Step 4: Subtract
\(A-A^{-1}=\begin{bmatrix}2+7&-3-3\\5+5&-7-2\end{bmatrix}=\begin{bmatrix}9&-6\\10&-9\end{bmatrix}\).
Final Answer:
A minus A inverse equals the matrix with rows (9, -6) and (10, -9). \[ \boxed{\text{(B) }\begin{bmatrix} 9 & -6 \\ 10 & -9 \end{bmatrix}} \]
For a 2 by 2 matrix \(\begin{bmatrix}a&b\\c&d\end{bmatrix}\), the inverse is \(\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}\).
Step 2: Find the determinant
\(|A|=2(-7)-(-3)(5)=-14+15=1\).
Step 3: Find the inverse
\(A^{-1}=\begin{bmatrix}-7&3\\-5&2\end{bmatrix}\).
Step 4: Subtract
\(A-A^{-1}=\begin{bmatrix}2+7&-3-3\\5+5&-7-2\end{bmatrix}=\begin{bmatrix}9&-6\\10&-9\end{bmatrix}\).
Final Answer:
A minus A inverse equals the matrix with rows (9, -6) and (10, -9). \[ \boxed{\text{(B) }\begin{bmatrix} 9 & -6 \\ 10 & -9 \end{bmatrix}} \]
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