Question

If P = `[(1, 2),(2, -1)]` and Q = `[(1, 0),(2, 1)]`, then compute:

1. P² – Q²
2. (P + Q)(P – Q)

Is (P + Q)(P – Q) = P² – Q² true for matrix algebra?

Answer 1

P² = `[(1, 2),(2, -1)][(1, 2),(2, -1)]`

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

= `[(5, 0),(0, 5)]`

Q² = `[(1, 0),(2, 1)][(1, 0),(2, 1)]`

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

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

i. P² – Q² = `[(5, 0),(0, 5)] - [(1, 0),(4, 1)]`

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

P + Q = `[(1, 2),(2,-1)] + [(1, 0),(2, 1)]`

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

P – Q = `[(1, 2),(2, -1)] - [(1, 0),(2, 1)]`

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

ii. (P + Q)(P – Q) = `[(2,2),(4, 0)][(0, 2),(0, -2)]`

= `[(0 + 0, 4 - 4),(0 + 0, 8 - 0)]`

= `[(0, 0),(0, 8)]`

Clearly, it can be said that (P + Q)(P – Q) = P² – Q² not true for matrix algebra.