Question

If \(A^{-1}=\frac{1}{11}\begin{pmatrix}-3 & 4\\5 & -3\end{pmatrix}\), then \(A=\)

• A. $\frac{-1}{11}\begin{pmatrix}3&4
  5&3\end{pmatrix}$
• B. $\frac{1}{11}\begin{pmatrix}3&4
  5&3\end{pmatrix}$
• C. $\begin{pmatrix}3&-4
  -5&3\end{pmatrix}$
• D. $\begin{pmatrix}3&4
  5&3\end{pmatrix}$
• E. $\begin{pmatrix}-3&4
  5&-3\end{pmatrix}$

Hint:

Verification Tip: You can easily check your work by multiplying your answer by the original inverse. \(A \times A^{-1}\) must equal the Identity Matrix \(I = \begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}\).

Answer 1

The Correct Option is D

Concept:
A fundamental property of inverse matrices is that the inverse of an inverse returns the original matrix: \((A^{-1})^{-1} = A\). The formula for the inverse of a \(2 \times 2\) matrix \(M = \begin{pmatrix}a & b\\ c & d\end{pmatrix}\) is \(M^{-1} = \frac{1}{|M|} \begin{pmatrix}d & -b\\ -c & a\end{pmatrix}\), where \(|M| = ad - bc\).

Step 1: Distribute the scalar into the inverse matrix.
To avoid errors, bring the scalar \(\frac{1}{11}\) inside the matrix \(A^{-1}\) before computing its inverse: \[ A^{-1} = \begin{pmatrix} \frac{-3}{11} & \frac{4}{11}\\ \frac{5}{11} & \frac{-3}{11} \end{pmatrix} \]

Step 2: Calculate the determinant of \(A^{-1}\).
Using \(|M| = ad - bc\): \[ |A^{-1}| = \left(\frac{-3}{11}\right)\left(\frac{-3}{11}\right) - \left(\frac{4}{11}\right)\left(\frac{5}{11}\right) \] \[ |A^{-1}| = \frac{9}{121} - \frac{20}{121} = \frac{-11}{121} = \frac{-1}{11} \]

Step 3: Find the adjoint of \(A^{-1}\).
Swap diagonal elements and change signs of off-diagonal elements: \[ \text{Adj}(A^{-1}) = \begin{pmatrix} \frac{-3}{11} & \frac{-4}{11}\\ \frac{-5}{11} & \frac{-3}{11} \end{pmatrix} \]

Step 4: Apply the inverse formula to find \(A\).
Since \(A = (A^{-1})^{-1} = \frac{1}{|A^{-1}|} \text{Adj}(A^{-1})\): \[ A = \frac{1}{\left(\frac{-1}{11}\right)} \begin{pmatrix} \frac{-3}{11} & \frac{-4}{11}\\ \frac{-5}{11} & \frac{-3}{11} \end{pmatrix} \] \[ A = -11 \begin{pmatrix} \frac{-3}{11} & \frac{-4}{11}\\ \frac{-5}{11} & \frac{-3}{11} \end{pmatrix} \]

Step 5: Distribute the scalar to simplify.
\[ A = \begin{pmatrix} 3 & 4\\ 5 & 3 \end{pmatrix} \] Hence the correct answer is (D) \(\begin{pmatrix}3 & 4\\ 5 & 3\end{pmatrix}\).