Question

If $A = \begin{bmatrix}1&2\\ -4&-1\end{bmatrix}$ then $A^{-1}$ is

• A. $\frac{1}{7} \begin{bmatrix}-1&-2\\ 4&1\end{bmatrix} $
• B. $\frac{1}{7} \begin{bmatrix}1&2\\ -4&-1\end{bmatrix} $
• C. $\frac{1}{7} \begin{bmatrix}-1&2\\ 4&-1\end{bmatrix} $
• D. Does not exist

Answer 1

Answer: (A)

$|A| = - 1 + 8 = 7$
$\text{adj} \left(A\right) = \begin{bmatrix}+\left(-1\right)&-\left(2\right)\\ -\left(-4\right)&+\left(1\right)\end{bmatrix} = \begin{bmatrix}-1&-2\\ 4&1\end{bmatrix}$
$\because A^{-1}=\frac{1}{|A|} \text{Adj} A$
$A^{-1} = \frac{1}{7} \begin{bmatrix}-1&-2\\ 4&1\end{bmatrix} $