Question

Solve the following :

If A = `[(1, 2, 3),(2, 4, 6),(1, 2, 3)],"B" = [(1, -1, 1),(-3, 2, -1),(-2, 1, 0)]`, then show that AB and BA are bothh singular martices.

Answer 1

AB = `[(1, 2, 3),(2, 4, 6),(1, 2, 3)] [(1, -1, 1),(-3, 2, -1),(-2, 1, 0)]`

= `[(1 - 6 - 6, -1 + 4 + 3, 1 - 2 + 0),(2 - 12 - 12, -2 + 8 + 6, 2 - 4 + 0),(1 - 6 - 6, -1 + 4 + 3, 1 - 2 + 0)]`

= `[(-11, 6, -1),(-22, 12, -2),(-11, 6, -1)]`

∴ |AB| = `|(-11, 6, -1),(-22, 12, -2),(-11, 6, -1)|`

= 0    ...[∵ R₁ an R₃ are identical]
∴ AB is a singular matrix.

BA = `[(1, -1, 1),(-3, 2, -1),(-2, 1, 0)][(1, 2, 3),(2, 4, 6),(1, 2, 3)]`

= `[(1 - 2 + 1, 2 - 4 + 2, 3 - 6 + 3),(-3 + 4 - 1, -6 + 8 - 2, -9 + 12 - 3),(-2 + 2 + 0, -4 + 4 + 0, -6 + 6 + 0)]`

= `[(0, 0, 0),(0, 0, 0),(0, 0, 0)]`

∴ |BA| = 0
∴ BA is a singular matrix.