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If A is a symmetric matrix and B is a sk...

If A is a symmetric matrix and B is a skew-symmetric matrix such that `A + B = [{:(2,3),(5,-1):}]`, then AB is equal to

A

`[{:(-4,-2),(-1,4):}]`

B

`[{:(4,-2),(-1,-4):}]`

C

`[{:(4,-2),(1,-4):}]`

D

`[{:(-4,2),(1,4):}]`

Text Solution

Verified by Experts

The correct Answer is:
B
Given matrix A is a symmetric and matrix B is a skew-symmetric.
`A^(T) = A " and " B^(T) = -B`
`"Since " A + B = [{:(2, 3), (5, -1):}] " " ("given")….(i)`
On taking transpose both sides, we get
`(A +B)^(T) = [{:(2, 3), (5, -1):}]^(T)`
`rArr A^(T) +B^(T) = [{:(2, 5), (3, -1):}] " " ..... (ii)`
Given, `A^(T) = A " and " B^(T) = -B`
`rArr A-B = [{:(2, 5), (3, -1):}]`
On solving Eqs. (i) and (ii), we get
` A= [{:(2, 4), (4, -1):}] " and " B = [{:(0, -1), (1, 0):}]`
So, ` AB= [{:(2, 4), (4, -1):}] [{:(0, -1), (1, 0):}] = [{:(4, -2), (-1, -4):}]`
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