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If A is a 3 $\times$ 3 matrix such that $\left | 5. adjA \right |=5$ , then $\left | A \right |$ is equal to :

Option 1)
$\pm \: \frac{1}{5}$

Option 2)
$\pm \: 5$

Option 3)
$\pm \: 1$

Option 4)
$\pm \: \frac{1}{25}$

Answers (2)

As we learnt in

Property of adjoint of A -

$\left | adj A \right |=\left | A \right |^{n-1}$

- wherein

$adj A$ denotes adjoint of $A$ and $\left |A \right |$ denotes determinant of $A$ and $n$ is the order of the matrix

$\left | adj\left ( A \right ) \right |=\left | A \right |^{n-1}$

$\left | K\:adj\left ( A \right ) \right |=K^{n}\left | adj\left ( A \right ) \right |$

So that, put n = 3

$\therefore \left | 5\:adj\left ( A \right ) \right |= 125\left | adj\left ( A \right ) \right |$

$\therefore 125\left | adj\left ( A \right ) \right |=5$

$\therefore \left | adj\left ( A \right ) \right |= \frac{1}{25}$

But $\left | adj\left ( A \right ) \right |= \left | A \right |^{2}$

$\therefore \left | A \right |^{2}= \frac{1}{25}$

$\left | A \right |= \pm \frac{1}{5}$

Option 1)

$\pm \: \frac{1}{5}$

This option is correct.

Option 2)

$\pm \: 5$

This option is incorrect.

Option 3)

$\pm \: 1$

This option is incorrect.

Option 4)

$\pm \: \frac{1}{25}$

This option is incorrect.

Posted by

Sabhrant Ambastha

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