Q.Express the matrix $\begin{pmatrix}1 & 5\\-1 & 2\end{pmatrix}$ as the sum of a symmetric matrix and a skew-symmetric matrix. OR If $A=\begin{pmatrix}3 & 1\\-1 & 2\end{pmatrix}$, then show that $A^2-5A+7I=0$. Hence find $A^{-1}$.

Matrix Decomposition (Symmetric + Skew-Symmetric)

At Class-12 level, "matrix decomposition" refers to a neat and useful theorem: every square matrix can be written as the sum of a symmetric matrix and a skew-symmetric matrix, and this split is unique.

The theorem

For any square matrix $A$,

where $A'$ is the transpose of $A$.

Why each part is what we claim

Write $P = \tfrac12(A+A')$ and $Q = \tfrac12(A-A')$. Then clearly $P + Q = A$. Now transpose each:

So the two halves genuinely are a symmetric matrix and a skew-symmetric matrix that add back to $A$.

How to use it — a worked method

Take $A = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}$. First find $A' = \begin{bmatrix} 2 & 6 \\ 4 & 8 \end{bmatrix}$.

Indeed $P + Q = A$, with $P$ symmetric and $Q$ skew-symmetric. …