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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

#### 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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