Let A = [ 0, 1, 0; 1, 0, 0; 0, 0, 1 ]. Then the number of 3 × 3 matrices B with entries from the set 1, 2, 3, 4, 5 and satisfying AB = BA is ____________.

Solution

Maths sir · 4 steps

This A just swaps things: rows of B in AB, columns of B in BA, so match the two swaps entry by entry.

1. What A does. Take B with entries a, b, c in row 1, d, e, f in row 2 and g, h, i in row 3. Multiplying on the left by A swaps rows 1 and 2 of B. Multiplying on the right by A swaps columns 1 and 2 of B.

   AB = (d, e, f; a, b, c; g, h, i), BA = (b, a, c; e, d, f; h, g, i)

2. Match entries. Setting AB = BA entry by entry gives only four distinct conditions: e = a, d = b, f = c and h = g. The entry i matches itself, so it has no condition.
3. Count free entries. So B has the form with rows (a, b, c), (b, a, c) and (g, g, i). The five entries a, b, c, g, i can each be chosen freely from the set, which gives 5 choices for each one.
4. So. The number of matrices is 5⁵ = 3125.

The idea to keep. Before you multiply a permutation matrix out, ask what it does: on the left it moves rows, on the right it moves columns. Then count only the entries that remain free.

Answer from the NTA final answer key. Solution by PrepWiser.