Q.Sum of two skew symmetric matrices is always _________ matrix.

The key idea is that a skew-symmetric matrix $A$ satisfies $A^T = -A$.

Step 1: Let $A$ and $B$ be two skew-symmetric matrices of the same order. Then $A^T = -A$ and $B^T = -B$.

Step 2: Consider their sum $C = A + B$. Take the transpose:

$C^T = (A + B)^T = A^T + B^T = (-A) + (-B) = -(A + B) = -C$. …

Why This Works: The Core Idea

The property of being skew-symmetric is about what happens when you transpose the matrix. If $A$ is skew-symmetric, flipping it across the main diagonal gives you the original matrix with every sign flipped. The question asks: if you take two matrices that each obey this rule, does their sum still obey it?

The answer lies in how transposition interacts with addition. Transpose is a linear operation — the transpose of a sum is simply the sum of the transposes. This single fact makes the proof almost immediate.

Step-by-Step Reasoning

1. State what we know about each matrix.

Let $A$ and $B$ be two skew-symmetric matrices of the same order (say $n \times n$). By definition:

2. Consider the sum $A + B$ and take its transpose.

We want to check whether $A + B$ is skew-symmetric. Start by transposing the sum:

3. Use the property that transpose distributes over addition.

This is the crucial step. For any two matrices of the same order:

Method: Show a class of matrices is closed under addition

Use this to prove that the sum of matrices sharing a property (here, skew-symmetric) keeps that property.

Steps

Step 1: State the defining property for each matrix.

For skew-symmetric $A$ and $B$: $A' = -A$ and $B' = -B$.

Step 2: Transpose the sum using linearity.

Transpose distributes over addition:

$(A+B)' = A' + B'.$ …

Common Mistakes

Mistake 1: Answering "symmetric".

Why it's wrong: transposing the sum gives $-(A+B)$, not $(A+B)$, so the sum is skew-symmetric, not symmetric. Correct approach: apply the transpose test to $A+B$ before naming the type.

Mistake 2: Assuming the result could depend on the specific entries. …