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Model Paper 3 — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSModel Paper 35 Marks
Question
Express the matrix $B=\left[\begin{array}{ccc}2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3\end{array}\right]$ as the sum of a symmetric and a skew-symmetric matrix.
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Answer

$B^{\prime}=\left[\begin{array}{ccc}2 & -1 & 1 \\ -2 & 3 & -2 \\ -4 & 4 & -3\end{array}\right]$
Let $P=\frac{1}{2}\left(B+B^{\prime}\right)=\left[\begin{array}{ccc}2 & \frac{-3}{2} & \frac{-3}{2} \\ \frac{-3}{2} & 3 & 1 \\ \frac{-3}{2} & 1 & -3\end{array}\right]$
$P^{\prime}=\left[\begin{array}{ccc}2 & \frac{-3}{2} & \frac{-3}{2} \\ \frac{-3}{2} & 3 & 1 \\ \frac{-3}{2} & 1 & -3\end{array}\right]=P$
Thus $P=\frac{1}{2}\left(B+B^{\prime}\right)$ is a symmetric matrix
Let $Q=\frac{1}{3}\left(B-B^{\prime}\right)=\left[\begin{array}{ccc}0 & \frac{-1}{2} & \frac{-5}{2} \\ \frac{1}{2} & 0 & 3 \\ \frac{5}{2} & -3 & 0\end{array}\right]$
$\begin{array}{l}Q^{\prime}=\left[\begin{array}{ccc}0 & \frac{-1}{2} & \frac{5}{2} \\ \frac{-1}{2} & 0 & -3 \\ \frac{-5}{2} & 3 & 0\end{array}\right] \\ Q^{\prime}=\left[\begin{array}{ccc}0 & \frac{-1}{2} & \frac{-5}{2} \\ \frac{1}{2} & 0 & 3 \\ \frac{5}{2} & -3 & 0\end{array}\right]\end{array}$
$Q^{\prime}=-Q$
Thus $Q=\frac{1}{2}\left(B-B^{\prime}\right)$ is a skew symmetric matrix
$P+Q=\left[\begin{array}{ccc}2 & \frac{-3}{2} & \frac{-3}{2} \\ \frac{-3}{2} & 3 & 1 \\ \frac{-3}{2} & 1 & -3\end{array}\right]+\left[\begin{array}{ccc}0 & \frac{-1}{2} & \frac{-5}{2} \\ \frac{1}{2} & 0 & 3 \\ \frac{5}{2} & -3 & 0\end{array}\right]$

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