Q.If $A$ and $B$ are symmetric matrices, prove that $AB - BA$ is a skew symmetric matrix.

Concept: Symmetric and Skew-Symmetric Matrices

A matrix $P$ is symmetric if $P^T = P$, and skew-symmetric if $P^T = -P$.

Step 1: Given $A$ and $B$ are symmetric, so $A^T = A$ and $B^T = B$.

Step 2: Consider $(AB - BA)^T$. Using the transpose property $(XY)^T = Y^T X^T$, we get:

Step 3: Substitute $A^T = A$ and $B^T = B$:

Since the transpose equals the negative of the original matrix, $AB - BA$ is skew-symmetric.

The key here is to understand what symmetric and skew-symmetric matrices mean in terms of transposes. A symmetric matrix equals its own transpose: $A^T = A$. A skew-symmetric matrix equals the negative of its transpose: $C^T = -C$.

When you multiply symmetric matrices, the product $AB$ is not necessarily symmetric — but its transpose has a neat property: $(AB)^T = B^T A^T = BA$ (since $A^T = A$ and $B^T = B$). This is the engine that drives the proof.

Let’s walk through it step by step.

1. Start with the transpose of the expression.

   We want to check if $AB - BA$ is skew-symmetric. That means we need to compute $(AB - BA)^T$ and see if it equals $-(AB - BA)$.

2. Use the transpose of a sum/difference.

   The transpose of a sum is the sum of transposes:

3. Apply the product rule for transposes. Remember: $(XY)^T = Y^T X^T$. So:

4. Substitute the symmetry condition. Since $A$ and $B$ are symmetric, $A^T = A$ and $B^T = B$. This gives:

5. Put it together.

6. Factor out a negative sign. Notice that $BA - AB = -(AB - BA)$. Therefore:

This is exactly the definition of a skew-symmetric matrix: a matrix $C$ such that $C^T = -C$.

Method: Proving a matrix is skew symmetric via its transpose

To prove an expression is skew symmetric, show its transpose equals its own negative, using the reversal law $(XY)'=Y'X'$ together with any given symmetry.

Steps

Step 1: Transpose the expression

$(AB-BA)'=(AB)'-(BA)'=B'A'-A'B'$.

Step 2: Substitute the given properties

With $A'=A$ and $B'=B$ (both symmetric), this becomes $BA-AB$.

Step 3: Show it is the negative of the original

$BA-AB=-(AB-BA)$, i.e. $(AB-BA)'=-(AB-BA)$, the definition of skew symmetric.

Common Mistakes

Mistake 1: Forgetting to reverse the order in $(AB)'$

Why it's wrong: $(AB)'=B'A'$, not $A'B'$; using the wrong order breaks the proof. Correct approach: apply the reversal law, transposed factors in the opposite order.

Mistake 2: Assuming $AB$ itself is symmetric

Why it's wrong: symmetry of $A$ and $B$ does not make $AB$ symmetric unless they commute; the result relies on transposing the whole difference. Correct approach: work with $(AB-BA)'$ as a single unit.