If A is an 3 × 3 non- singular matrix, such that AA^ ' = A^ ' A and...

A

is an

3 \times 3

non- singular matrix, such that

AA^{'} = A^{'}A

and

B = A^{- 1}A^{'}

, then

BB^{'}

equals to

Options

A $\left(B^{- 1}\right) '$
B $I + B$
C $I$
D $B^{- 1}$

Correct answer

C. $I$

Step-by-step solution

Given,

$AA^{'} = A^{'}A$

and $B = A^{- 1}A^{'}$

$BB^{'} = \left(A^{- 1}A^{'}\right) \left(A^{- 1}A^{'}\right) '$

$= \left(A^{- 1}A^{'}\right)\left[ \left(A^{'}\right) '\left(A^{- 1}\right) '\right]$

$= \left(A^{- 1}A^{'}\right) \left[A \left(A^{'}\right)- 1\right]$

\left[∵ \left(A^{'}\right)' = A, \left(A^{- 1}\right)' = \left(A^{'}\right)- 1\right] $</p><p>$ = A^{- 1} \left(A^{'}A\right) \left(A^{'}\right) $- 1$

$= A^{- 1}\left(AA^{'}\left(A^{'}\right)- 1\right)</p><p>$ = \left(A^{- 1}A\right) \left(A^{- 1}A\right) $'$

$= I \cdotI = I$

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