Q.Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

Concept: Every square matrix $A$ can be uniquely written as $A = P + Q$, where $P = \frac{1}{2}(A + A^T)$ is symmetric and $Q = \frac{1}{2}(A - A^T)$ is skew-symmetric.

Steps for each matrix:

1. Compute $A^T$.
2. Find $P = \frac{1}{2}(A + A^T)$.
3. Find $Q = \frac{1}{2}(A - A^T)$.
4. Verify $A = P + Q$.

(i) $A = \begin{bmatrix} 3 & 5 \\ 1 & -1 \end{bmatrix}$, $A^T = \begin{bmatrix} 3 & 1 \\ 5 & -1 \end{bmatrix}$ …

The idea is beautiful in its simplicity. Any square matrix can be split into two parts: one that is symmetric (equal to its own transpose) and one that is skew-symmetric (equal to the negative of its transpose). The trick is to use the transpose itself to manufacture these parts.

If you take any matrix $A$, then $A + A^T$ is always symmetric — because transposing it gives itself back. Similarly, $A - A^T$ is always skew-symmetric — because transposing it flips its sign. Halving each gives the exact decomposition.

Let's apply this to each matrix.

(i) $A = \begin{bmatrix} 3 & 5 \\ 1 & -1 \end{bmatrix}$

Step 1: Find $A^T$.

Transpose means swap rows and columns:

Step 2: Compute $P = \frac{1}{2}(A + A^T)$.

Add element-wise:

Now halve:

Check: $P$ is symmetric — $P_{12} = P_{21} = 3$.

Step 3: Compute $Q = \frac{1}{2}(A - A^T)$.

Subtract:

Halve:

Check: $Q$ is skew-symmetric — diagonal entries are zero, and $Q_{12} = -Q_{21}$.

Step 4: Verify. …

Method: Expressing a square matrix as a symmetric plus a skew-symmetric matrix

Every square matrix decomposes uniquely as $A=P+Q$ with $P=\tfrac12(A+A')$ symmetric and $Q=\tfrac12(A-A')$ skew symmetric.

Steps

Step 1: Compute $A'$

Interchange rows and columns.

Step 2: Build $P$ and $Q$

$P=\tfrac12(A+A'),\qquad Q=\tfrac12(A-A').$ …

Common Mistakes

Mistake 1: Dropping the factor $\tfrac12$ in $P$ and $Q$

Why it's wrong: without halving, $P+Q$ becomes $2A$, not $A$; the $\tfrac12$ is essential to the decomposition. Correct approach: always divide $A\pm A'$ by 2.

Mistake 2: Not verifying $P+Q=A$ and the symmetry/skew properties …