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.

The key idea is that any square matrix can be uniquely written as the sum of a symmetric matrix $S$ and a skew-symmetric matrix $K$, where

$S = \frac{1}{2}(B + B^T)$ and $K = \frac{1}{2}(B - B^T)$.

Step 1: Compute $B^T$.

$B^T = \begin{bmatrix} 2 & -1 & 1 \\ -2 & 3 & -2 \\ -4 & 4 & -3 \end{bmatrix}$

Step 2: Find $S = \frac{1}{2}(B + B^T)$.

$B + B^T = \begin{bmatrix} 4 & -3 & -3 \\ -3 & 6 & 2 \\ -3 & 2 & -6 \end{bmatrix}$

$S = \frac{1}{2}(B + B^T) = \begin{bmatrix} 2 & -1.5 & -1.5 \\ -1.5 & 3 & 1 \\ -1.5 & 1 & -3 \end{bmatrix}$

Step 3: Find $K = \frac{1}{2}(B - B^T)$. …

The Core Idea: Every Square Matrix Has a Built-in Mirror

This problem is about matrix decomposition — breaking a matrix into two special pieces that reveal hidden structure. Every square matrix $A$ can be written as:

Why does this always work? Because any matrix $A$ can be "averaged" with its own transpose. The symmetric part is the average of $A$ and $A^T$; the skew-symmetric part is half their difference. This is analogous to writing any function as the sum of an even and an odd function — a deep mathematical symmetry.

Step-by-Step Construction

1. Find the transpose $B^T$

The transpose swaps rows and columns. For $B = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}$, we get:

Notice how the diagonal stays the same (2, 3, -3) — this is always true for the transpose.

2. Compute the symmetric part $P = \frac{1}{2}(B + B^T)$

Add corresponding entries:

Now divide by 2:

Check: $P$ is symmetric — $P_{12} = P_{21} = -\frac{3}{2}$, $P_{13} = P_{31} = -\frac{3}{2}$, $P_{23} = P_{32} = 1$. Good.

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

Subtract entry by entry:

Divide by 2:

$Q = \frac{1}{2} \begin{bmatrix} 0 & -1 & -5 \\ 1 & 0 & 6 \\ 5 & -6 & 0 \end{bmatrix} = \begin{bmatrix} 0 & -\frac{1}{2} & -\frac{5}{2} \\ \frac{1}{2} & 0 & 3 \\ \frac{5}{2} & -3 & 0 \end{bmatrix}$ …

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

Use this whenever asked to write a square matrix as the sum of a symmetric and a skew-symmetric matrix.

Steps

Step 1: Compute the transpose $A'$.

Swap rows and columns of the given matrix.

Step 2: Form the two halves. …

Common Mistakes

Mistake 1: Forgetting the $\tfrac12$ factor.

Why it's wrong: $A + A'$ is symmetric but equals $2P$; using it directly doubles the symmetric part. Correct approach: halve both $A+A'$ and $A-A'$.

Mistake 2: Swapping the two formulas.

Why it's wrong: the sum $\tfrac12(A+A')$ is symmetric and the difference $\tfrac12(A-A')$ is skew-symmetric — not the other way round. Correct approach: sum $\to$ symmetric, difference $\to$ skew. …