Q.If $A$ and $B$ are symmetric matrices of same order, then $AB$ is symmetric if and only if _________.

Concept: Symmetric and Skew-Symmetric Matrices — a product of two symmetric matrices is symmetric only when they commute.

Step 1: For symmetric $A$ and $B$, we have $A^T = A$ and $B^T = B$.

Step 2: The transpose of the product is $(AB)^T = B^T A^T = BA$. …

Why This Approach Works

The problem asks about the condition under which the product of two symmetric matrices remains symmetric. This is a classic result in matrix theory that hinges on a single property: the transpose of a product reverses the order of multiplication. When both $A$ and $B$ are symmetric, $A^T = A$ and $B^T = B$. So $(AB)^T = B^T A^T = BA$. For $AB$ to be symmetric, we need $(AB)^T = AB$, which forces $BA = AB$. That is the entire story — the condition is commutativity.

Many students mistakenly think the product of symmetric matrices is always symmetric. It is not. Only when they commute does the symmetry survive.

Step-by-Step Reasoning

1. State what is given. $A$ and $B$ are symmetric matrices of the same order. That means:

2. Write the condition for $AB$ to be symmetric. By definition, $AB$ is symmetric if and only if:

3. Apply the transpose rule. The transpose of a product reverses the order: $(AB)^T = B^T A^T.$ …

Method: Find the condition for a product of symmetric matrices to be symmetric

Use this to state when $AB$ is symmetric, given $A,B$ symmetric.

Steps

Step 1: Write the symmetry condition you must satisfy.

$AB$ is symmetric exactly when $(AB)' = AB$.

Step 2: Expand the transpose with the reversal law. …

Common Mistakes

Mistake 1: Believing the product of two symmetric matrices is always symmetric.

Why it's wrong: it is symmetric only when the matrices commute ($AB=BA$). Correct approach: state the commuting condition explicitly.

Mistake 2: Writing $(AB)' = A'B'$ rather than $B'A'$. …