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प्रश्न
If A = `[(-1, 2, 1),(-3, 2, -3)]` and B = `[(2, 1),(-3, 2),(-1, 3)]`, prove that (A + BT)T = AT + B.
उत्तर
A = `[(-1, 2, 1),(-3, 2, -3)] "and B" = [(2, 1),(-3, 2),(-1, 3)]`
∴ AT = `[(-1, -3),(2, 2),(1, -3)] "and B"^"T" = [(2, -3, -1),(1, 2, 3)]`
∴ A + BT = `[(-1, 2, 1),(-3, 2, -3)] + [(2, -3, 1),(1, 2, 3)]`
= `[(-1 + 2, 2 - 3, 1 - 1),(-3 + 1, 2 + 2, -3 + 3)]`
= `[(1, -1, 0),(-2, 4, 0)]`
∴ (A + BT)T = `[(1, -2),(-1, 4),(0, 0)]` ...(i)
Now, AT + B = `[(-1, -3),(2, 2),(1, -3)] + [(2, 1),(-3, 2),(-1, 3)]`
= `[(-1 + 2, -3 + 1),(2 - 3, 2 + 2),(1 - 1, -3 + 3)]`
= `[(1, -2),(-1, 4),(0, 0)]` ...(ii)
From (i) and (ii), we get
(A + BT)T = AT + B.
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