# Solve the following : If A = [2-33-2-14],B=[-3412-1-3], verify (A + 2BT)T = AT + 2B. - Mathematics and Statistics

Sum

Solve the following :

If A = [(2, -3),(3, -2),(-1, 4)],"B" = [(-3, 4, 1),(2, -1, -3)], verify (A + 2BT)T = AT + 2B.

#### Solution

A = [(2, -3),(3, -2),(-1, 4)] "and B" = [(-3, 4, 1),(2, -1, -3)]

∴ AT = [(2, 3, -1),(-3, -2, 4)] "and B"^"T" = [(-3, 2),(4, -1),(1, -3)]

∴ A + 2BT = [(2, -3),(3, -2),(-1, 4)] + 2[(-3, 2),(4, -1),(1, -3)]

= [(2, -3),(3, -2),(-1, 4)] + [(-6, 4),(8, -2),(2, -6)]

= [(2 - 6, -3 + 4),(3 + 8 , -2 - 2),(-1 - 2, 4 - 6)]

∴ A + 2BT = [(-4, 1),(11, -4),(1, -2)]

∴ *A + 2BT)T = [(-4, 11, 1),(1, -4, -2)]     ...(i)

A + 2B = [(2, 3, -1),(-3, -2, 4)] + 2[(-3, 4, 1),(2, -1, -3)]

= [(2, 3, -1),(-3, -2, 4)] + [(-6, 8, 2),(4, -2, -6)]

= [(-4, 11, 1),(1, -4, -2)]      ...(iii)
From (i) and (ii), we get
(A + 2BT)T = AT + 2B.

Concept: Properties of Matrices
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Chapter 2: Matrices - Miscellaneous Exercise 2 [Page 84]

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