Express $A=\begin{pmatrix}3&5\1&-2\end{pmatrix}$ as the sum of a symmetric matrix and a skew-symmetric matrix.

Question

Accepted Answer

First find the transpose: $A'=\begin{pmatrix}3&1\5&-2\end{pmatrix}$.

**Symmetric part:**
$$	frac12(A+A')=	frac12\left[\begin{pmatrix}3&5\1&-2\end{pmatrix}+\begin{pmatrix}3&1\5&-2\end{pmatrix}ight]=	frac12\begin{pmatrix}6&6\6&-4\end{pmatrix}=\begin{pmatrix}3&3\3&-2\end{pmatrix}$$
Call this $P$. Check $P'=P$: transposing $\begin{pmatrix}3&3\3&-2\end{pmatrix}$ swaps the off-diagonal $3$'s with each other, giving back the same matrix — so $P$ is genuinely symmetric.

**Skew-symmetric part:**
$$	frac12(A-A')=	frac12\left[\begin{pmatrix}3&5\1&-2\end{pmatrix}-\begin{pmatrix}3&1\5&-2\end{pmatrix}ight]=	frac12\begin{pmatrix}0&4\-4&0\end{pmatrix}=\begin{pmatrix}0&2\-2&0\end{pmatrix}$$
Call this $Q$. Check $Q'=-Q$: transposing $\begin{pmatrix}0&2\-2&0\end{pmatrix}$ gives $\begin{pmatrix}0&-2\2&0\end{pmatrix}$, which is exactly $-Q$ — confirming $Q$ is genuinely skew-symmetric, and its diagonal entries are $0$ as required.

**Independent check:** add $P$ and $Q$ back together — $\begin{pmatrix}3&3\3&-2\end{pmatrix}+\begin{pmatrix}0&2\-2&0\end{pmatrix}=\begin{pmatrix}3&5\1&-2\end{pmatrix}=A$ ✓ — recovering the original matrix exactly, which independently confirms both parts were computed correctly.

> [!ANSWER]
> $A=\begin{pmatrix}3&3\3&-2\end{pmatrix}+\begin{pmatrix}0&2\-2&0\end{pmatrix}$ (symmetric $+$ skew-symmetric).

Q.Express $A=\begin{pmatrix}3&5\\1&-2\end{pmatrix}$ as the sum of a symmetric matrix and a skew-symmetric matrix.

Concept understanding — Algebra of Matrices

Q.Express A=(351−2)A=\begin{pmatrix}3&5\\1&-2\end{pmatrix}A=(31​5−2​) as the sum of a symmetric matrix and a skew-symmetric matrix.

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Q.Express $A=\begin{pmatrix}3&5\\1&-2\end{pmatrix}$ as the sum of a symmetric matrix and a skew-symmetric matrix.

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