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GATE 2020 ME (ME2) – Question 12

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigen values and eigen vectors · 1 mark · Multiple choice

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

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

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Correct answer: (B) $\begin{pmatrix}-4&2&5\\6&3&7\\-1&0&2\end{pmatrix}$

Explanation

By definition $S=(P+P^T)/2$ and $V=(P-P^T)/2$, so P=S+V. Add entrywise to obtain $P=\begin{pmatrix}-4&2&5\\6&3&7\\-1&0&2\end{pmatrix}$. This is **B**. S−V would instead reconstruct Pᵀ.