Question

If X = `[(4 , 1),(-1 , 2)]`, show that 6X - X² = 9I, where I is unit matrix.

Answer 1

Here
X² = X·X
= `[(4 , 1),(-1 , 2)]·[(4 , 1),(-1 , 2)]`
= `[(16 -1, 4 + 2),(-4 -2, -1 + 4)] = [(15 , 6),(-6 , 3)]`
L.H.S. = 6X - X²
= `6[(4 , 1),(-1 , 2)] - [(15 , 6),(-6 , 3)]`
= `[(24 , 6),(-6 , 12)] - [(15 , 6),(-6 , 3)]`
= `[(24 - 15, 6 - 6),(-6 + 6 , 12 - 3)]`
= `[(9 , 0),(0 ,9)]`
= `9[(1 , 0),(0 , 1)]`
= 9I = R.H.S.
Hence proved.