If $A=\left[\begin{array}{ccc}2 & 0 & -3 \ 4 & 3 & 1 \ -5 & 7 & 2\end{array} ight]$ is expressed as a sun of a symmetric matrix $\mathrm{P}$ and skew symmetric matrix $\mathrm{Q}$, then $\mathrm{P}^{\mathrm{T}}-\mathrm{Q}^{\mathrm{T}}=$

Question

If $A=\left[\begin{array}{ccc}2 & 0 & -3 \ 4 & 3 & 1 \ -5 & 7 & 2\end{array}ight]$ is expressed as a sun of a symmetric matrix $\mathrm{P}$ and skew symmetric matrix $\mathrm{Q}$, then $\mathrm{P}^{\mathrm{T}}-\mathrm{Q}^{\mathrm{T}}=$

Aicharya · Accepted Answer

Given : $A=\left[\begin{array}{ccc}2 & 0 & -3 \ 4 & 3 & 1 \ -5 & 7 & 2\end{array}ight]$ Since, each and every matrix can be written as sum of symmetric \& skew symmetric matrix. $	herefore$ A can be written as sum symmetric matrix $P$ and skew symmetric matrix Q. where $\mathrm{P}=\frac{1}{2}\left[\mathrm{~A}+\mathrm{A}^{\mathrm{T}}ight]$ $ \mathrm{Q}=\frac{1}{2}\left[\mathrm{~A}-\mathrm{A}^{\mathrm{T}}ight] $ $ \begin{aligned} & 	ext { Now, } A^T=\left[\begin{array}{ccc} 2 & 4 & -5 \ 0 & 3 & 7 \ -3 & 1 & 2 \end{array}ight] \ & 	herefore P=\frac{1}{2}\left[\begin{array}{ccc} 4 & 4 & -8 \ 4 & 6 & 8 \ -8 & 8 & 4 \end{array}ight]=\left[\begin{array}{ccc} 2 & 2 & -4 \ 2 & 3 & 4 \ -4 & 4 & 2 \end{array}ight] \ & Q=\frac{1}{2}\left[\begin{array}{ccc} 0 & -4 & 2 \ 4 & 0 & -6 \ -2 & 6 & 0 \end{array}ight]=\left[\begin{array}{ccc} 0 & -2 & 1 \ 2 & 0 & -3 \ -1 & 3 & 0 \end{array}ight] \ & \mathrm{P}^T=\left[\begin{array}{ccc} 2 & 2 & -4 \ 2 & 3 & 4 \ -4 & 4 & 2 \end{array}ight] Q^T=\left[\begin{array}{ccc} 0 & 2 & -1 \ -2 & 0 & +3 \ 1 & -3 & 0 \end{array}ight] \end{aligned} $ $\mathrm{P}^{\mathrm{T}}-\mathrm{Q}^{\mathrm{T}}=\left[\begin{array}{ccc}2-0 & 2-2 & -4+1 \ 2+2 & 3-0 & 4-3 \ -4-1 & 4+3 & 2-0\end{array}ight]=\left[\begin{array}{ccc}2 & 0 & -3 \ 4 & 3 & 1 \ -5 & 7 & 2\end{array}ight]$

If $A=\left[\begin{array}{ccc}2 & 0 & -3 \\ 4 & 3 & 1 \\ -5 & 7 & 2\end{array}\right]$ is expressed as a sun…

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