If A and B Are Symmetric Matrices of the Same Order, Show That AB Is Symmetric if and Only if AB = BA

NCERT Class 12 Mathematics — Matrices, Example 24.

Problem

If $A$ and $B$ are symmetric matrices of the same order, show that $AB$ is symmetric if and only if $AB=BA$.

Key insight. “Symmetric” simply means a matrix equals its own transpose. Since $(AB)^T = B^TA^T$ for any two matrices, substituting the fact that $A$ and $B$ are each already equal to their own transposes turns $(AB)^T$ directly into $BA$ — connecting the transpose of the product to the reversed product almost automatically.

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Approach

Use the standard identity $(AB)^T=B^TA^T$, then substitute $A^T=A$ and $B^T=B$ (since both are given as symmetric). This shows $(AB)^T=BA$ directly. Since “$AB$ is symmetric” means precisely “$(AB)^T=AB$”, substituting the expression just found turns this condition into $BA=AB$ — proving the “if and only if” in both directions at once.

Solution

Step 1 — Write the transpose of the product AB

For any two matrices (of compatible sizes) $A$ and $B$:

$$(AB)^T = B^TA^T$$

Step 2 — Use the fact that A and B are symmetric

Since $A$ and $B$ are symmetric, $A^T=A$ and $B^T=B$. Substituting:

$$(AB)^T = BA$$

Step 3 — Apply the definition of “symmetric” to AB

By definition, $AB$ is symmetric if and only if $(AB)^T=AB$.

Step 4 — Combine Steps 2 and 3

Substituting the result from Step 2 into the condition from Step 3:

$$AB \text{ is symmetric} \iff (AB)^T=AB \iff BA=AB$$

Answer

Common mistakes

• Forgetting that $(AB)^T=B^TA^T$ reverses the order of the product. Writing $(AB)^T=A^TB^T$ instead (without reversing) breaks the entire proof, since it would incorrectly suggest $(AB)^T=AB$ always, regardless of whether $A$ and $B$ commute.
• Only proving one direction of the “if and only if.” Since the derivation in Step 4 is a chain of equivalences (each $\iff$), both directions are proven simultaneously — but it’s worth explicitly noting that the argument works both ways, rather than treating it as a one-directional implication.

Practise next

• If $A$ and $B$ are symmetric matrices of the same order, show that $AB-BA$ is always a skew-symmetric matrix, using a similar transpose-based argument.