If A, B are two n n non - singular matrices, then - Vidyadip

MCQ

If $A, B$ are two $n \times n$ non $-$ singular matrices, then

✓
$AB$ is non $-$ singular.
B
$AB$ is singular.
C
$(A B)^{-1} A^{-1} B^{-1}$
D
$(A B)^{-1}$ does not exist.

✓

Answer

Correct option: A.

$AB$ is non $-$ singular.

$A$ and $B$ are non-singular matrices of order $n \times n.$
$\therefore|\text{A}|\neq0$ and $|\text{B}|\neq0\ .....(\text{i})$
$A$ and $B$ are of the same order, so $AB$ is defined and is of the same order.
Thus,
$|AB| = |A\|B|$
$\Rightarrow|\text{AB}|\neq0\ \big[$Using $(1)\big]$
Thus, $Ab$ is non $-$ singular.

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