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Let $A, B, C$ be $3 \times 3$ non-singular matrices and $I$ be the identity matrix of order three. If $A B A=B A^2 B$ and $A^3=I$, then $A B^4-B^4 A=$
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Verified Answer
The correct answer is:
$\mathrm{O}_{3 \times 3}$
Given, $A B A=B A^2 B$
and
$A^3=I$
$\Rightarrow$ $A B A=B A^2 B$
$\Rightarrow$ $A B A A^2=B A^2 B A^2$
$\Rightarrow$ $A B A^3=B A^2 B A^2$
$\Rightarrow$ $A B=B A^2 B A^2$
$\Rightarrow \quad A B^2=B A^2 B A^2 B$
$\Rightarrow \quad A B^2=B A^2 A B A$
$\Rightarrow \quad A B^2=B A^3 B A$
$\Rightarrow \quad A B^2=B I B A$
$\Rightarrow \quad A B^4=B^2 B^2 A$
$\Rightarrow \quad A B^4=B^4 A$
$A B^4-B^4 A=0$
and
$A^3=I$
$\Rightarrow$ $A B A=B A^2 B$
$\Rightarrow$ $A B A A^2=B A^2 B A^2$
$\Rightarrow$ $A B A^3=B A^2 B A^2$
$\Rightarrow$ $A B=B A^2 B A^2$
$\Rightarrow \quad A B^2=B A^2 B A^2 B$
$\Rightarrow \quad A B^2=B A^2 A B A$
$\Rightarrow \quad A B^2=B A^3 B A$
$\Rightarrow \quad A B^2=B I B A$
$\Rightarrow \quad A B^4=B^2 B^2 A$
$\Rightarrow \quad A B^4=B^4 A$
$A B^4-B^4 A=0$
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