If A is a symmetric matrix and B is a skew symmetric matrix such that A+B=235-1 then A B is equal to

A+B=[2 35 −1]−−−−(1)AT+BT=[2 53 −1]A−B=[2 53 −1] −−−−(2)∴ by solvingA=[2 44 −1],B=[ 0 −1−1 0]∴A.B=[ 4 −2−1 −4]

If A is a symmetric matrix and B is a skew symmetric matrix such that A+B=235-1 then A B is equal to

A+B=[2 35 −1]−−−−(1)AT+BT=[2 53 −1]A−B=[2 53 −1] −−−−(2)∴ by solvingA=[2 44 −1],B=[ 0 −1−1 0]∴A.B=[ 4 −2−1 −4]

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If A is a symmetric matrix and B is a skew symmetric matrix such that $A + B = [\begin{matrix} 2 & 3 \\ 5 & - 1 \end{matrix}]$ then A B is equal to

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$[\begin{matrix} - 4 & 2 \\ 1 & 4 \end{matrix}]$

$[\begin{matrix} - 4 & - 2 \\ - 1 & 4 \end{matrix}]$

$[\begin{matrix} 4 & - 2 \\ - 1 & - 4 \end{matrix}]$

$[\begin{array}{l} 4 - 2 \\ 1 - 4 \end{array}]$

answer is C.

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Detailed Solution

$A + B = [\begin{array}{l} 2 3 \\ 5 - 1 \end{array}] - - - - (1)$
$\begin{array}{l} A^{T} + B^{T} = [\begin{array}{l} 2 5 \\ 3 - 1 \end{array}] \\ A - B = [\begin{array}{l} 2 5 \\ 3 - 1 \end{array}] - - - - (2) \end{array}$
$∴$ by solving
$\begin{array}{l} A = [\begin{array}{l} 2 4 \\ 4 - 1 \end{array}], B = [\begin{array}{l} 0 - 1 \\ - 1 0 \end{array}] \\ ∴ A . B = [\begin{array}{l} 4 - 2 \\ - 1 - 4 \end{array}] \end{array}$