Question

A = `[(3 , 1),(-1 , 2)]`, show that A² - 5A + 7 I₂ = 0.

Answer 1

We have
A = `[(3 , 1),(-1 , 2)]`
A = `[(3 , 1),(-1 , 2)][(3 , 1),(-1 , 2)]`
= `[(9 - 1, 3 + 2),(-3 -2, -1 + 4)]`
= `[(8, 5),(-5, 3)]`
-5A = `[((-5)·3 ,(-5)·1),((-5)·(-1),(-5)·2)]`
= `[(-5 , -5),(5 , -10)]`
7I₂ = `7[(1 , 0),(0 , 1)] = [(7 , 0),(0 , 7)]`
So A² - 5A + 71₂
= `[(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)]`
So A² - 5A + 7I₂ = 0.
Hence proved.