Let A be a non singular matrix of order $3 \times 3$ such that det $A = 5$ and B is a non-singular matrix such that $A^{- 1} B^{2} + A B = 0$ . If the value of $| A^{6} - 2 A^{4} B + A^{2} B^{2} | = N,$ then $\log_{10} N =$

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answer is 6.

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Detailed Solution

$A^{- 1} B^{2} = - A B B^{2} = - A^{2} B \Rightarrow B = - A^{2} A^{6} - 2 A^{4} B + A^{2} B^{2} = 4 A^{6} | 4 A^{6} | = 4^{3} {| A |}^{6} = 4^{3} 5^{6} = 2^{6} {.5}^{6} = 10^{6}$

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