Question

If A = `[(4, 2),(-1, 1)]`, show that (A − 2I) (A − 3J) = 0.

Answer 1

A = `[(4, 2),(-1, 1)]`

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

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

A − 2I = `[(4, 2),(-1, 1)] - [(2, 0),(0, 2)]`

`[(4 - 2, 2),(-1, 1 - 2)]`

= `[(2, 2),(-1, -1)]`

A − 3J = `[(4, 2),(-1, 1)] - [(0, 3),(3, 0)]`

= ` [(4, 2−3),(−1−3, 1)]`

= `[(4, -1),(-4, 1)]`

(A − 2I) (A − 3J) = `[(2, 2),(-1, -1)] xx [(4, -1),(-4, 1)]`

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

= `[(8 - 8, -2 + 2),(-4 + 4, 1 - 1)]`

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

So, (A − 2I) (A − 3J) = 0