Question

If M = `[(4,1),(-1,2)]`, show that 6M – M² = 9I; where I is a 2 × 2 unit matrix.

Answer 1

M² = `[(4, 1),(-1, 2)][(4, 1),(-1, 2)]`

= `[(4 xx 4 + 1 xx (-1), 4 xx 1 + 1 xx 2),(-1 xx 4 + 2 xx (-1), -1 xx 1 + 2 xx 2)]`

= `[(16 - 1,4 + 2),(-4 - 2, -1 + 4)]`

= `[(15, 6),(-6, 3)]`

6M – M² = `6[(4, 1),(-1, 2)] - [(15, 6),(-6, 3)]`

= `[(24, 6),(-6,12)] - [(15,6),(-6,3)]`

= `[(24 - 15, 6 - 6),(-6 - (-6), 12 - 3)]`

= `[(9, 0),(0, 9)]`

= `9[(1, 0),(0, 1)]`

= 9I

Hence proved.