1. Tardigrade
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  3. Mathematics
  4. If A=[6 8 5 4 2 3 9 7 1] is the sum of a symmetric matrix B and skew-symmetric matrix C, then B is equal to

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Q. If $A=\begin{bmatrix}6 & 8 & 5 \\ 4 & 2 & 3 \\ 9 & 7 & 1\end{bmatrix}$ is the sum of a symmetric matrix $B$ and skew-symmetric matrix $C$, then $B$ is equal to

Matrices

Solution:

We have, $A=\begin{bmatrix}6 & 8 & 5 \\ 4 & 2 & 3 \\ 9 & 7 & 1\end{bmatrix} \Rightarrow A^{\prime}=\begin{bmatrix}6 & 4 & 9 \\ 8 & 2 & 7 \\ 5 & 3 & 1\end{bmatrix}$
$\because \quad A=\left(\frac{A+A^{\prime}}{2}\right)+\left(\frac{A-A^{\prime}}{2}\right)$
Here, $\frac{A+A^{\prime}}{2}$ is the symmetric part of $A$ and $\frac{A-A^{\prime}}{2}$ is the skew-symmetric part of $A$.
$\therefore$ We have, symmetric matrix, $B=\frac{A+A^{\prime}}{2}$
$=\frac{1}{2}\left\{\begin{bmatrix}6 & 8 & 5 \\ 4 & 2 & 3 \\ 9 & 7 & 1\end{bmatrix}+\begin{bmatrix}6 & 4 & 9 \\ 8 & 2 & 7 \\ 5 & 3 & 1\end{bmatrix}\right\}$
$=\frac{1}{2}\begin{bmatrix}12 & 4 & 10 \\ 14 & 10 & 2\end{bmatrix}=\begin{bmatrix}6 & 2 & 5 \\ 7 & 5 & 1\end{bmatrix}$