10. (25 points)

a. Determine.

(i) (1 points) True (T) or false (F): If $A$ is an invertible matrix, then $\det(A^{-1}) = [\det(A)]^{-1}$.

(ii) (1 points) True (T) or false (F): If $M$ is an $n \times n$ matrix and can be written in the form

$M = \begin{pmatrix} A & B \\ O & C \end{pmatrix},$

where $A, B, C, O$ are square matrices, and $O$ is all zero matrix. Then $\det(M) = \det(A) \cdot \det(C)$.

(iii) (3 points) Find the determinant of the following matrix:

$\begin{bmatrix} 2 & 0 & -1 & 1 & 3 \\ 0 & -1 & 0 & -2 & 0 \\ 2 & 1 & 4 & 0 & 3 \\ 2 & -1 & -1 & -3 & 3 \\ 0 & 2 & 0 & 3 & 1 \end{bmatrix}.$