Question

If A = `[(6, 5),(7, 6)]`, show that A² − 12A + I = 0.

Answer 1

A = `[(6, 5),(7, 6)]`

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

A² = `[(6, 5),(7, 6)] xx [(6, 5),(7, 6)]`

= `[(6 xx 6 + 5 xx 7, 6 xx 5 + 5 xx 6),(7 xx 6 + 6 xx 7, 7 xx 5 + 6 xx 6)]`

= `[(36 + 35, 30 + 30),(42 + 42, 35 + 36)]`

= `[(71, 60),(84, 71)]`

12A = 12 × `[(6, 5),(7, 6)]`

= `[(72, 60),(84, 72)]`

A² − 12A = `[(71, 60),(84, 71)] − [(72, 60),(84, 72)]`

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

A² − 12A + I = `[(-1, 0),(0, -1)] + [(1, 0),(0, 1)]`

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

= 0

A² − 12A + I = 0

Hence proved.