If \(A = \begin{bmatrix}2&3\\1&2\end{bmatrix}\), then \(A + A^{-1}\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\begin{bmatrix}4&0\\0&4\end{bmatrix}\)
For a \(2 \times 2\) matrix \(\begin{bmatrix}a&b\\c&d\end{bmatrix}\) with \(ad - bc \ne 0\), the inverse is
\( \displaystyle \frac{1}{ad - bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix} \)
Here
\(ad - bc = 2(2) - 3(1) = 1\)
\( \displaystyle A^{-1} = \begin{bmatrix}2&-3\\-1&2\end{bmatrix} \)
Adding:
\( \displaystyle A + A^{-1} = \begin{bmatrix}2 + 2&3 - 3\\1 - 1&2 + 2\end{bmatrix} = \begin{bmatrix}4&0\\0&4\end{bmatrix} \)