Question:
If $A, B$ are two $n \times n$ non-singular matrices, then
(a) $A B$ is non-singular
(b) $A B$ is singular
(c) $(A B)^{-1} A^{-1} B^{-1}$
(d) $(A B)^{-1}$ does not exist
Solution:
(a) $A B$ is non-singular
$A$ and $B$ are non-singular matrices of order $n \times n$.
$\therefore|\mathrm{A}| \neq 0$ and $|\mathrm{B}| \neq 0 \quad \ldots(1)$
$A$ and $B$ are of the same order, so $A B$ is defined and is of the same order.
Thus,
$|\mathrm{AB}|=|\mathrm{A}||\mathrm{B}|$
$\Rightarrow|\mathrm{AB}| \neq 0 \quad[$ Using $(1)]$
Thus, $A B$ is non-singular.
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