Question

Show that `[(1, 2),(2, 1)]` is a solution of the matrix equation X² – 2X – 3I = 0, where I is the unit matrix of order 2.

Answer 1

Given

x² – 2x – 3I = 0

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

∴ x² = `[(1, 2),(2, 1)][(1, 2),(2, 1)]`

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

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

= `[(5, 4),(4, 5)]`

x² − 2x − 3I = `[(5, 4),(4, 5)] - 2[(1, 2),(2, 1)] - 3[(1, 0),(0, 1)]`

= `[(5, 4),(4, 5)] - [(2, 4),(4, 2)] - [(3, 0),(0, 3)]`

= `[(5 - 2 - 3, 4 - 4 + 0),(4 - 4 - 0, 5 - 2 - 3)]`

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

∴ x² = 2x – 3I = 0

Hence proved.