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If `P = [(1, 2),(2, -1)] and Q = [(1, 0),(2, 1)]` Then Compute: 1) `P^2 - Q^2` 2) (P + Q)(P - Q) `Is (P + Q)(P - Q) = P^2 - Q^2` True for Matrix Algebra? - ICSE Class 10 - Mathematics

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Question

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

1) `P^2 - Q^2`

2) (P + Q)(P - Q)

`Is (P + Q)(P - Q) = P^2 - Q^2` true for matrix algebra?

Solution

`P^2 = [(1, 2),(2, -1)][(1, 2),(2, -1)]= [(1+ 4, 2-2),(2 - 2, 4 + 1)] = [(5, 0),(0, 5)]`

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

`p^2 - Q^2 = [(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)]`

`(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) = P^2 - Q^2` not true for matrix algebra.

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Solution If `P = [(1, 2),(2, -1)] and Q = [(1, 0),(2, 1)]` Then Compute: 1) `P^2 - Q^2` 2) (P + Q)(P - Q) `Is (P + Q)(P - Q) = P^2 - Q^2` True for Matrix Algebra? Concept: Matrices Examples.
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