Let \(A = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}\). Then the number of \(3 \times 3\) matrices \(B\) with entries from the set \(\{1, 2, 3, 4, 5\}\) and satisfying \(AB = BA\) is ____________.
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Correct Answer: 3125
Solution and Explanation
\(AB\) interchanges rows 1 and 2 of \(B\): \(AB = \begin{bmatrix} d & e & f \\ a & b & c \\ g & h & i \end{bmatrix}\).
\(BA\) interchanges columns 1 and 2 of \(B\): \(BA = \begin{bmatrix} b & a & c \\ e & d & f \\ h & g & i \end{bmatrix}\).
\(AB = BA\) gives \(d = b,\ e = a,\ f = c,\ g = h\), and \(i\) is unrestricted.
Free entries: \(a, b, c, g, i\), i.e. 5 independent entries, each from 5 values.
Number of matrices \(= 5^5 = 3125\).
Answer: 3125
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