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

The key idea is that a skew-symmetric matrix satisfies $A^T = -A$. Squaring it reveals the nature of $A^2$.

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

$(A^2)^T = (A A)^T = A^T A^T = (-A)(-A) = A^2.$ …

Why This Works: The Core Idea

A skew-symmetric matrix is defined by the property that its transpose equals its negative: $A^T = -A$. This means that for any entry $a_{ij}$, we have $a_{ij} = -a_{ji}$, and importantly, all diagonal entries must be zero (since $a_{ii} = -a_{ii}$ implies $a_{ii} = 0$).

When you square such a matrix, something interesting happens to its symmetry. The transpose of a product reverses the order: $(AB)^T = B^T A^T$. So for $A^2$, we have $(A^2)^T = (A \cdot A)^T = A^T A^T$. But since $A^T = -A$, this becomes $(-A)(-A) = A^2$. The two negatives cancel out, leaving the original matrix unchanged. That is the signature of a symmetric matrix.

Let's walk through this step by step.

1. Start with the definition.

   A matrix $A$ is skew-symmetric if $A^T = -A$. This is the only fact we need — no special properties of $A$ beyond this.

2. Consider $A^2$ and take its transpose.

   We want to check whether $A^2$ is symmetric (equal to its transpose) or skew-symmetric (equal to the negative of its transpose). So compute:

Using the rule $(AB)^T = B^T A^T$, we get:

3. Substitute the skew-symmetry condition. Since $A^T = -A$, replace each $A^T$:

4. Interpret the result. …

Method: Find the symmetry type of the square of a matrix

Use this to determine what $A^2$ becomes when $A$ is symmetric or skew-symmetric.

Steps

Step 1: State the given symmetry.

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

Step 2: Transpose $A^2$ with the reversal law.

$(A^2)' = (A\cdot A)' = A' A' .$ …

Common Mistakes

Mistake 1: Assuming $A^2$ stays skew-symmetric like $A$.

Why it's wrong: $(A^2)' = (-A)(-A) = A^2$, so the two negatives cancel and $A^2$ is symmetric, not skew-symmetric. Correct approach: carry the signs through the product.

Mistake 2: Forgetting to reverse the order when transposing $A\cdot A$. …