If A and B are 3 × 3 matrices and | A | ne 0 , then which of following are true?

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3 and 4 .Determinants and Matrices

hard

If $A$ and $B$ are $3 × 3$ matrices and $| A | \ne 0$, then which of the following are true?

A

$| AB | = 0 ==> | B | = 0$

B

$| AB | = 0 ==> B = 0$

C

$| A^{-1} | = | A |^{-1}$

D

both $(A)$ and $(C)$

Solution

For $ | AB | = 0 ==> | A | · | B | = 0$

==> $| A | = 0, | B | = 0$

$AA^{-1} = I$ ==> $| A | · | A |^{-1} = | I | = 1$

==> $| A^{-1} | =$ $\frac{1}{{|A|}}$ $= | A |^{-1}$

Standard 12

Mathematics

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