Question

If \( A=\begin{bmatrix}2 & 3 \\ 1 & 2\end{bmatrix} \) and \( B=\begin{bmatrix}2 & -3 \\-1 & 2\end{bmatrix} \), then \( (B^{-1}A^{-1})^{-1} \) is

• A. \( \begin{bmatrix}2&3\\1&-2\end{bmatrix} \)
• B. \( \begin{bmatrix}0&1\\1&0\end{bmatrix} \)
• C. \( \begin{bmatrix}1&2\\3&4\end{bmatrix} \)
• D. \( \begin{bmatrix}1&0\\0&1\end{bmatrix} \)

Hint:

Always remember: \( (AB)^{-1} = B^{-1}A^{-1} \).

Answer 1

The Correct Option is D

Step 1: Use the inverse property.
For matrices, \[ (B^{-1}A^{-1})^{-1} = A B \] 

Step 2: Multiply matrices \( A \) and \( B \). 
\[ AB = \begin{bmatrix}2&3\\1&2\end{bmatrix} \begin{bmatrix}2&-3\\-1&2\end{bmatrix} \] \[ = \begin{bmatrix} 4-3 & -6+6 \\ 2-2 & -3+4 \end{bmatrix} = \begin{bmatrix}1&0\\0&1\end{bmatrix} \] 

Step 3: Conclusion. 
\[ \boxed{\begin{bmatrix}1&0\\0&1\end{bmatrix}} \]