Q.If $A$ is skew symmetric matrix, then $A^2$ is a symmetric matrix.

The key idea is that a skew-symmetric matrix satisfies $A^T = -A$, and we examine the transpose of $A^2$.

Step 1: Take the transpose of $A^2$:

Step 2: Since $A$ is skew-symmetric, $A^T = -A$. Substitute:

$A^T A^T = (-A)(-A) = A^2$ …

Why This Works: The Core Idea

A skew-symmetric matrix is one where flipping it across the main diagonal (transposing) gives back the negative of the original. In symbols: $A^T = -A$. Think of it as a matrix that "anti-reflects" — every entry $a_{ij}$ is the negative of $a_{ji}$, and the diagonal entries are forced to be zero.

Now, when you square such a matrix, something interesting happens. The "negative sign" gets multiplied by itself, and two negatives make a positive. That's the heart of why $A^2$ ends up symmetric — the skew-symmetry gets "squared away."

Step-by-Step Proof

1. Start with the definition of skew-symmetry.

We are given that $A$ is skew-symmetric. By definition:

This is the only fact we need — no special properties of $A$ beyond this.

2. Write what we want to prove.

We need to show $A^2$ is symmetric. A matrix $M$ is symmetric if $M^T = M$. So we need to check:

3. Use the transpose property for a product.

A fundamental rule: the transpose of a product is the product of the transposes in reverse order. That is:

Applying this to $A^2 = A \cdot A$:

4. Substitute the skew-symmetry condition.

Since $A^T = -A$, we replace each $A^T$:

$(A^2)^T = (-A)(-A) = (-1)(-1) \cdot A \cdot A = A^2$ …

Method: Finding the Symmetry Type of a Power of a Skew-Symmetric Matrix

Use this to classify powers (or products) of symmetric / skew-symmetric matrices. Transpose the expression and track the sign that the skew condition contributes.

Steps

Step 1: Encode the hypothesis.

$A$ skew-symmetric means $A'=-A$.

Step 2: Transpose the target using the reversal law.

For $A^2=A\cdot A$:

Step 3: Substitute the skew condition and read the sign. …

Common Mistakes

Mistake 1: Losing track of the sign from $A'=-A$.

Why it's wrong: forgetting that $(-A)(-A)=+A^2$ can lead to the wrong (skew) conclusion. Correct approach: carefully multiply the two negatives to get a positive.

Mistake 2: Ignoring order-reversal in the transpose of a product.

Why it's wrong: $(A^2)'=A'A'$ comes from $(XY)'=Y'X'$; skipping this rule is unsafe for general products. Correct approach: always apply the reversal law, even when both factors are the same. …