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  4. A and B are two square matrices such that A2 B=B A and if (A B)10=Ak B10, then k is

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Q. $A$ and $B$ are two square matrices such that $A^{2} B=B A$ and if $(A B)^{10}=A^{k} B^{10}$, then $k$ is

Matrices

Solution:

$(A B) \cdot(A B)=A(B A) B=A^{3} B^{2}$
$(A B)(A B)(A B)=A^{7} B^{3}$
so $(A B)^{ n }=A^{2^{n}-1} B^{n}$
so $k=2^{10}-1=1023$