Question:
If the matrix \(A=\begin{bmatrix} 3 & -5 \\ 4 & -7 \end{bmatrix}\) and \(A^{-1}=\begin{bmatrix} 7 & -5 \\ 4 & a \end{bmatrix}\), then the value of \(a\) is...
If the matrix \(A=\begin{bmatrix} 3 & -5 \\ 4 & -7 \end{bmatrix}\) and \(A^{-1}=\begin{bmatrix} 7 & -5 \\ 4 & a \end{bmatrix}\), then the value of \(a\) is...
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Use the adjoint formula for the inverse of a 2x2 matrix and compare the entries.
Updated On: Nov 12, 2026
- \(-3\)
- \(3\)
- \(5\)
- \(-5\)
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept
For \(A=\begin{bmatrix}a&b\\c&d\end{bmatrix}\), \(A^{-1}=\dfrac{1}{|A|}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}\).
Step 2: Determinant
\(|A|=3(-7)-(-5)(4)=-21+20=-1\).
Step 3: Find the inverse
\(A^{-1}=\dfrac{1}{-1}\begin{bmatrix}-7&5\\-4&3\end{bmatrix}=\begin{bmatrix}7&-5\\4&-3\end{bmatrix}\).
Step 4: Compare
Comparing with the given inverse, \(a=-3\).
Final Answer:
The value of \(a\) is \(-3\). \[ \boxed{\text{(A) }-3} \]
For \(A=\begin{bmatrix}a&b\\c&d\end{bmatrix}\), \(A^{-1}=\dfrac{1}{|A|}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}\).
Step 2: Determinant
\(|A|=3(-7)-(-5)(4)=-21+20=-1\).
Step 3: Find the inverse
\(A^{-1}=\dfrac{1}{-1}\begin{bmatrix}-7&5\\-4&3\end{bmatrix}=\begin{bmatrix}7&-5\\4&-3\end{bmatrix}\).
Step 4: Compare
Comparing with the given inverse, \(a=-3\).
Final Answer:
The value of \(a\) is \(-3\). \[ \boxed{\text{(A) }-3} \]
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