Question

Find the matrix A, if `B = [(2, 1),(0, 1)]` and `B^2 = B + 1/2 A`.

Answer 1

`B^2 = B + 1/2 A`

`1/2 A = B^2 - B`

A = 2(B² – B)

`B^2 = [(2, 1),(0, 1)][(2, 1),(0, 1)]`

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

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

= `[(4, 3),(0, 1)]`

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

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

= `[(2, 2),(0, 0)]`

∴ A = 2(B² – B)

= `2[(2, 2),(0, 0)]`

= `[(4, 4),(0, 0)]`