Q.Express the matrix $\begin{bmatrix} 2 & 3 & 1 \\ 1 & -1 & 2 \\ 4 & 1 & 2 \end{bmatrix}$ as the sum of a symmetric and a skew symmetric matrix.

Concept: Symmetric And Skew Symmetric Matrices

Any square matrix $A$ can be uniquely written as $A = P + Q$, where

$P = \frac{1}{2}(A + A^T)$ is symmetric ($P^T = P$), and

$Q = \frac{1}{2}(A - A^T)$ is skew symmetric ($Q^T = -Q$).

Step 1: Find $A^T$ (transpose of $A$).

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

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

Why This Works: The Symmetric + Skew-Symmetric Decomposition

Any square matrix $A$ can be split into two special parts. The symmetric part is a matrix that equals its own transpose — it's "mirrored" across the main diagonal. The skew-symmetric part is a matrix whose transpose equals its negative — the off-diagonal entries are opposite in sign, and the diagonal entries are always zero.

The beauty is that you don't need to guess. There's a direct formula:

Why does this work? If you take the transpose of $\frac{1}{2}(A + A^T)$, you get $\frac{1}{2}(A^T + A)$, which is the same matrix — so it's symmetric. And the transpose of $\frac{1}{2}(A - A^T)$ is $\frac{1}{2}(A^T - A) = -\frac{1}{2}(A - A^T)$, which is exactly the condition for skew-symmetry.

Let's apply this to the given matrix.

Step 1: Write down the given matrix and find its transpose

We have:

The transpose $A^T$ is obtained by swapping rows and columns:

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

Add $A$ and $A^T$ entry by entry:

Now multiply each entry by $\frac{1}{2}$:

Notice that $P$ is symmetric — the entry at $(i,j)$ equals the entry at $(j,i)$. For example, $P_{12} = 2$ and $P_{21} = 2$.

Step 3: Compute the skew-symmetric part $Q = \frac{1}{2}(A - A^T)$

Subtract $A^T$ from $A$:

Now multiply by $\frac{1}{2}$:

$Q = \frac{1}{2}(A - A^T) = \begin{bmatrix} 0 & 1 & -1.5 \\ -1 & 0 & 0.5 \\ 1.5 & -0.5 & 0 \end{bmatrix}$ …

Method: Decomposing a square matrix into symmetric and skew-symmetric parts

Every square matrix can be written uniquely as the sum of a symmetric matrix and a skew-symmetric matrix. The construction uses the matrix and its transpose.

Steps

Step 1: Compute the transpose $A'$.

Swap rows and columns.

Step 2: Build the symmetric part.

$P$ is symmetric because $(A+A')'=A'+A=A+A'$.

Step 3: Build the skew-symmetric part.

$Q=\tfrac12(A-A').$ …

Common Mistakes

Mistake 1: Forgetting the factor of $\tfrac12$.

Why it's wrong: without it, $(A+A')+(A-A')=2A$, not $A$; the decomposition only works when each part is halved. Correct approach: use $P=\tfrac12(A+A')$ and $Q=\tfrac12(A-A')$.

Mistake 2: Swapping the roles — taking the symmetric part as $A-A'$.

Why it's wrong: $A+A'$ is symmetric and $A-A'$ is skew-symmetric; reversing them mislabels the parts. Correct approach: symmetric uses the sum, skew uses the difference. …