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प्रश्न
If A = `[(8, 4),(10, 5)], "B" = [(5, -4),(10, -8)]` show that (A + B)2 = A2 + AB + B2
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उत्तर
(A + B)2 = A2 + AB + B2
∴ (A + B)(A + B) = A2 + AB + B2
∴ A(A + B) + B(A + B) = A2 + AB + B2
∴ A2 + AB + BA + B2 = A2 + AB + B2
∴ `cancel("A"^2) + cancel("AB") + "BA" + cancel("B"^2) = cancel("A"^2) + cancel("AB") + cancel("B"^2)`
∴ BA = 0
We have to prove that
(A + B)2 = A2 + AB + B2,
i.e., to prove (A + B)(A + B) = A2 + AB + B2
i.e., to prove A2 + AB + BA + B2 = A2 + AB + B2,
i.e., to prove BA = 0.
BA = `[(5, -4),(10, -8)] [(8, 4),(10, 5)]`
= `[(40 - 40, 20 - 20),(80 - 80, 40 - 40)]`
= `[(0, 0),(0, 0)]`
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