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Properties of Determinants

If A , B , C ...

Question

If $A, B, C$ are square matrices of same order such that $A B = B A, C^{2} = B,$ then $(A^{- 1} C A)^{2}$ is equal to

A

$B^{2}$

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B

$A^{2}$

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C

$C^{2}$

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D

$C$

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Solution

The correct option is C
$C^{2}$

Given, $A B = B A, C^{2} = B$

$\Rightarrow (A^{- 1} C A)^{2} = (A^{- 1} C A) (A^{- 1} C A) = A^{- 1} C I C A = A^{- 1} C^{2} A$

$= A^{- 1} B A = A^{- 1} (A B) = I B = B = C^{2}$

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