Q.Express the matrix $B=\begin{bmatrix}2 & -2 & -4\\ -1 & 3 & 4\\ 1 & -2 & -3\end{bmatrix}$ as the sum of a symmetric and a skew symmetric matrix. OR By using elementary operations, find the inverse of the matrix $A=\begin{bmatrix}2 & 1\\ 1 & 1\end{bmatrix}$.

$B=P+Q$, with symmetric part $P$ and skew-symmetric part $Q$ as below.

Any square matrix $B$ can be written as $B=P+Q$ where $P=\dfrac{B+B'}{2}$ (symmetric) and $Q=\dfrac{B-B'}{2}$ (skew-symmetric).

$B=\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix},\qquad B'=\begin{bmatrix}2&-1&1\\-2&3&-2\\-4&4&-3\end{bmatrix}.$

$P=\frac{B+B'}{2}=\frac12\begin{bmatrix}4&-3&-3\\-3&6&2\\-3&2&-6\end{bmatrix}=\begin{bmatrix}2&-3/2&-3/2\\-3/2&3&1\\-3/2&1&-3\end{bmatrix}$

(check $P'=P$: symmetric)

$Q=\frac{B-B'}{2}=\frac12\begin{bmatrix}0&-1&-5\\1&0&6\\5&-6&0\end{bmatrix}=\begin{bmatrix}0&-1/2&-5/2\\1/2&0&3\\5/2&-3&0\end{bmatrix}$

(check $Q'=-Q$: skew-symmetric)

Verify $P+Q=B$: adding entrywise reproduces $B$ exactly.

OR: $A=\begin{bmatrix}2&1\\1&1\end{bmatrix}$. Write $A=IA$:

$\begin{bmatrix}2&1\\1&1\end{bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}A$

$R_1\leftrightarrow R_2$: $\begin{bmatrix}1&1\\2&1\end{bmatrix}=\begin{bmatrix}0&1\\1&0\end{bmatrix}A$

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