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  4. Let A and B are two non-singular matrices such that AB=BA2,B4=I and Ak=I, then k can be equal to

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Q. Let $A$ and $B$ are two non-singular matrices such that $AB=BA^{2},B^{4}=I$ and $A^{k}=I,$ then $k$ can be equal to

NTA AbhyasNTA Abhyas 2020Matrices

Solution:

$AB^{4}=AB\left(\right.B\left.\right)\left(\right.B\left.\right)\left(\right.B\left.\right)$
$=BA^{2}\left(\right.B\left.\right)\left(\right.B\left.\right)\left(\right.B\left.\right)$
$=BA\left(\right.AB\left.\right)BB$
$=BA\left(\right.BA^{2}\left.\right)BB$
$=B\left(\right.AB\left.\right)\left(\right.A^{2}BB\left.\right)$
$=BBA^{4}BB$
$=B^{2}BA^{8}B=B^{4}A^{16}$
$\Rightarrow A=A^{16}\Rightarrow A^{15}=I$