Question

Find A² – 5A + 6I, if A = `[(2, 0, 1),(2, 1, 3),(1, -1, 0)]`

Answer 1

We have A² = A × A

Given A = `[(2, 0, 1),(2, 1, 3),(1, -1, 0)]`

A² = A.A = `[(2,0,1),(2,1,3),(1, -1,0)] [(2,0,1),(2,1,3),(1, -1,0)]`

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

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

= `[(5, -1, 2), (9, -2, 5),(0, -1,-2)]`   ...(i)

5A = 5 = `[(2, 0, 1),(2, 1, 3),(1, -1,0)] = [(10, 0, 5),(10, 5, 15),(5, -5, 0)]`   ...(ii)

6I = 6 = `[(1, 0, 0),(0, 1, 0),(0, 0,1)] = [(6, 0, 0),(0, 6, 0),(0, 0, 0)]`   ...(iii)

A² – 5A + 6I = `[(5, -1, 2), (9, -2, 5),(0, -1, -2)] -  [(10, 0, 5),(10, 5, 15),(5, -5, 0)] + [(6, 0, 0),(0, 6, 0),(0, 0, 6)]`

= `[(-5, -1, -3), (-1, -7, -10),(-5, 4, -2)] + [(6, 0, 0),(0, 6, 0),(0, 0, 6)]`

= `[(1, -1, -3),(-1, -1, -10),(-5, 4, 4)]`