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Inverse of a Matrix

lf A is a 3...

Question

lf A is a $3 x 3$ non singular matrix and $| a d j A | = | A |^{x}, | a d j (a d j A) | = | A |^{y}, | A^{- 1} | = | A |^{z}$ then the values of $x, y, z$ , in descending order

A

X

,

Y

,

Z

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B

Z, Y, X

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C

Z, X, Y

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D

Y, X, Z

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Solution

The correct option is B $Y, X, Z$
$| a d j . A | = | A |^{n - 1}$
$| a d j . A | = | A |^{2} = | A |^{x}$
$∴ x = 2$
$| a d j (a d j A) | = | A |^{y} = | A |^{(3 - 1)^{2}}$
$y = 2^{2} = 4$
$| A^{- 1} | = | A |^{- 1} = | A |^{z}$
$z = - 1$
$∴ y > x > z$

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