MathematicsA and B are two square matrices such that A2B = BA and if (AB) 10 = AKB10, then k is

A and B are two square matrices such that A²B = BA and if (AB) ¹⁰ = A^KB¹⁰, then k is

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A
1001
B
1023
C
1042
D
none of these

Solution:

$\begin{array}{l} (AB) \cdot (AB) = A (BA) B \\ = A^{3} B^{2} \\ (AB) (AB) (AB) = A^{7} B^{3} \\ so (AB)^{n} = A^{2^{n} - 1} B^{n} \\ so k = 2^{10} - 1 = 1023 \end{array}$