What is the value of the determinant of the inverse of the matrix...

Let $A = \begin{pmatrix} -4 & -5 \\ 2 & 2 \end{pmatrix}$.

The determinant of matrix $A$ is given by:

$|A| = (-4)(2) - (-5)(2)$

$|A| = -8 + 10 = 2$

The determinant of the inverse of a matrix is the reciprocal of the determinant of the matrix, i.e., $|A^{-1}| = \dfrac{1}{|A|}$.

$|A^{-1}| = \dfrac{1}{2}$

Answer: $\dfrac{1}{2}$