If \(A = \begin{bmatrix}3 & -2\\ 4 & 1\end{bmatrix}\), then \(A(\mathrm{adj}\,A)\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\begin{bmatrix}11 & 0\\ 0 & 11\end{bmatrix}\)
For every square matrix, \(A(\mathrm{adj}\,A) = |A|\,I\). Here
\( |A| = 3 \times 1 - (-2) \times 4 = 3 + 8 = 11 \)
so
\( \displaystyle A(\mathrm{adj}\,A) = 11I = \begin{bmatrix}11 & 0\\ 0 & 11\end{bmatrix} \)
Check by direct product, with \(\mathrm{adj}\,A = \begin{bmatrix}1 & 2\\ -4 & 3\end{bmatrix}\):
\( \displaystyle \begin{bmatrix}3 & -2\\ 4 & 1\end{bmatrix}\begin{bmatrix}1 & 2\\ -4 & 3\end{bmatrix} = \begin{bmatrix}3 + 8 & 6 - 6\\ 4 - 4 & 8 + 3\end{bmatrix} = \begin{bmatrix}11 & 0\\ 0 & 11\end{bmatrix} \)