Question

Given matrix X = `[(1, 1),(8, 3)]` and I = `[(1, 0),(0, 1)]`, prove that X² = 4X + 5I.

Answer 1

Given X² = 4X + 5I

Solving for L.H.S.,

X² = `[(1, 1),(8, 3)][(1, 1),(8, 3)]`

= `[(1 xx 1 + (1) xx (8), 1 xx (1) + (1) xx 3),((8) xx 1 + 3 xx (8),(8) xx (1) + 3  xx 3)]`

= `[(1 + 8, 1 + 3),(8 + 24, 8 + 9)]`

= `[(9, 4),(32, 17)]`

Solving for R.H.S.,

4X + 5I = `4[(1, 1),(8, 3)] + 5[(1, 0),(0, 1)]`

= `[(4, 4),(32, 12)] + [(5, 0),(0, 5)]`

= `[(4 + 5, 4 + 0),(32 + 0, 12 + 4)]`

= `[(9, 4),(32, 17)]`

Since L.H.S. = R.H.S.

Hence proved.