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UPSC NDA Math Model Paper 8

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Question : 66 of 120

Marks: +1, -0

If

{A}

is a square matrix, then what is

{adj} {A}^{{T}}-({adj} {A})^{{T}}

equal to?

$2|A|$
$2| A|^T$
Null matrix
Unit matrix

Solution:

Given,

{A}

is a square matrix.

Consider,

({adj} {A}^{{T}})-({adj} {A})^{{T}}

=|{A}^{{T}}|({A}^{{T}})^{-1}-(|{A}| {A}^{-1})^{{T}}

=|{A}|^{{T}}({A}^{{T}})^{-1}-(|{A}|^{{T}}({A}^{-1})^{{T}}.

=|{A}|^{{T}}({A}^{-1})^{{T}}-(|{A}|^{{T}}({A}^{-1})^{{T}}.

=0

=

Null Matrix

Hence, If

{A}

is a square matrix, then what is

{adj} {A}^{{T}}-({adj} {A})^{{T}}

equal to null matrix.

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