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Question: If A = \(\begin{bmatrix} 2 & 0 & –3 \\ 4 & 3 & 1 \\ –5 & 7 & 2 \end{bmatrix}\) is expressed as the s...

If A = [20–3431–572]\begin{bmatrix} 2 & 0 & –3 \\ 4 & 3 & 1 \\ –5 & 7 & 2 \end{bmatrix} is expressed as the sum of a symmetric and skew symmetric matrix then the symmetric matrix is –

A

[24–5037–312]\begin{bmatrix} 2 & 4 & –5 \\ 0 & 3 & 7 \\ –3 & 1 & 2 \end{bmatrix}

B

[44–8468–884]\begin{bmatrix} 4 & 4 & –8 \\ 4 & 6 & 8 \\ –8 & 8 & 4 \end{bmatrix}

C

[22–4234–442]\begin{bmatrix} 2 & 2 & –4 \\ 2 & 3 & 4 \\ –4 & 4 & 2 \end{bmatrix}

D

[100010001]\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}

Answer

[22–4234–442]\begin{bmatrix} 2 & 2 & –4 \\ 2 & 3 & 4 \\ –4 & 4 & 2 \end{bmatrix}

Explanation

Solution

Sym matrix = A+AT2\frac{A + A^{T}}{2}

= [20–3431–572]\begin{bmatrix} 2 & 0 & –3 \\ 4 & 3 & 1 \\ –5 & 7 & 2 \end{bmatrix} + [24–5037–312]\begin{bmatrix} 2 & 4 & –5 \\ 0 & 3 & 7 \\ –3 & 1 & 2 \end{bmatrix} = [22–4234–442]\begin{bmatrix} 2 & 2 & –4 \\ 2 & 3 & 4 \\ –4 & 4 & 2 \end{bmatrix}