Let A, B, C be 3 times 3 non-singular matrice | vedclass

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(A) Given,$A B A = B A^2 B$ and $A^3 = I$. Multiplying both sides of $A B A = B A^2 B$ by $A^2$ on the right,we get: $A B A \cdot A^2 = B A^2 B \cdot

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$O_{3 	imes 3}$

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(A) Given,$A B A = B A^2 B$ and $A^3 = I$. Multiplying both sides of $A B A = B A^2 B$ by $A^2$ on the right,we get: $A B A \cdot A^2 = B A^2 B \cdot A^2$ $A B A^3 = B A^2 B A^2$ Since $A^3 = I$,we have: $A B = B A^2 B A^2$ Now,multiply by $B$ on the right: $A B^2 = B A^2 B A^2 B$ Substitute $A^2 B = A B A$ from the given equation: $A B^2 = B A^2 B (A B A)$ $A B^2 = B A^2 B A B A$ Using $A^3 = I$,we can simplify further. Alternatively,note that $A B A = B A^2 B \implies A B = B A^2 B A^{-1}$. Since $A^3 = I$,$A^{-1} = A^2$. Thus $A B = B A^2 B A^2$. By induction or repeated substitution,$A B^4 = B^4 A$. Therefore,$A B^4 - B^4 A = O_{3 	imes 3}$.

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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