Question

If A = `[(3, 1),(-1, 2)]`, prove that A² – 5A + 7I = 0, where I is a 2 x 2 unit matrix.

Answer 1

A² – 5A + 7I = A.A – 5A + 7I 

= `[(3, 1),(-1, 2)][(3, 1),(-1, 2)] 5[(3, 1),(-1, 2)] + 7[(1, 0),(0, 1)]`

= `[(9 - 1, 3 + 2),(-3 - 2, -1 + 4)]  [(15, 5),(-5, 10)] + [(7, 0),(0, 7)]`

= `[(8, 5),(-5, 3)] - [(15, 5),(-5, 10)] + [(7, 0),(0, 7)]`

= `[(8 - 15 + 7, 5 - 5 + 0),(-5 + 5 + 0, 3 - 10 + 7)]`

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

= 0.