GATE2020-ME-2: 2

Question

A matrix $P$ is decomposed into its symmetric part $S$ and skew symmetric part $V$. If $$S= \begin{pmatrix} -4 & 4 & 2 \\ 4 & 3 & 7/2 \\ 2 & 7/2 & 2 \end{pmatrix}, \: \: V= \begin{pmatrix} 0 & -2 & 3  \\ 2 & 0 & 7/2 \\ -3  & -7/2 & 0 \end{pmatrix}$$

then matrix $P$ is

• A. $\begin{pmatrix} -4 & 6 & -1  \\ 2 & 3 & 0 \\ 5  & 7 & 2 \end{pmatrix} \\$
• B. $\begin{pmatrix} -4 & 2 & 5  \\ 6 & 3 & 7 \\ -1  & 0 & 2 \end{pmatrix} \\$
• C. $\begin{pmatrix} 4 & -6 & 1  \\ -2 & -3 & 0 \\ -5  & -7 & -2 \end{pmatrix} \\$
• D. $\begin{pmatrix} -2 & 9/2 & -1  \\ -1 & 81/4 & 11 \\ -2  & 45/2 & 73/4 \end{pmatrix}$

Answer 1

We know that any square matrix  $M$  can be expressed as sum of symmetric and skew-symmetric matrices,

i.e.    $M=\frac{1}{2}(M+M^{T})+\frac{1}{2}(M-M^{T})$,   where the first and the second term on the RHS indicate the symmetric and skew-symmetric matrices respectively. 

(Refer: https://math.stackexchange.com/questions/875873/prove-square-matrix-can-be-written-as-a-sum-of-a-symmetric-and-skew-symmetric-m )

So, for the given question, option B turns out to be the only one satisfying the above equation.

Correct Option: B