If $A$ and $B$ are symmetric matrices, then (i) $AB - BA$ is a _________. (ii) $BA - 2AB$ is a _________.

> [!TLDR] > For symmetric matrices $A$ and $B$, the commutator $AB - BA$ is always skew-symmetric, while $BA - 2AB$ is neither symmetric nor skew-symmetric in general — it is simply an arbitrary matrix. ## Why This Approach Works The key idea is simple: a symmetric matrix equals its own transpose, and a skew-symmetric matrix equals the negative of its transpose. When you multiply symmetric matrices, the transpose of the product reverses the order: $(AB)^T = B^T A^T = BA$ (since $A^T = A$, $B^T = B$). So the transpose of $AB$ is $BA$, not $AB$ itself. This reversal is the engine behind the entire problem. We are asked to classify two expressions: $AB - BA$ and $BA - 2AB$. The first is a classic commutator — its behaviour under transpose is well-known. The second is a linear combination that does not have a standard name. We will check each by taking its transpose and comparing with the original. > [!WARNING] > A common mistake is to assume $AB$ is symmetric just because $A$ and $B$ are. That is false unless $A$ and $B$ commute. Always remember: $(AB)^T = B^T A^T$, not $A^T B^T$. --- ## Step-by-Step Solution ### 1. Recall the definitions A matrix $M$ is **symmetric** if $M^T = M$. A matrix $M$ is **skew-symmetric** if $M^T = -M$. Given: $A^T = A$ and $B^T = B$. ### 2. Transpose of a product For any matrices of compatible sizes: $$ (AB)^T = B^T A^T $$ Since $A$ and $B$ are symmetric, this becomes: $$ (AB)^T = BA $$ > [!TIP] > This is the only fact you need. The transpose flips the order, and symmetry removes the transposes on the individual matrices. ### 3. Analyse $AB - BA$ Let $C = AB - BA$. Take its transpose: $$ C^T = (AB - BA)^T = (AB)^T - (BA)^T = BA - AB $$ But $BA - AB = -(AB - BA) = -C$. So: $$ C^T = -C $$ This is exactly the definition of a skew-symmetric matrix. > [!FORMULA] > For symmetric $A$ and $B$, the commutator $AB - BA$ is always skew-symmetric: > $$ > (AB - BA)^T = -(AB - BA) > $$ Thus, the first blank is **skew-symmetric**. ### 4

If A and B are symmetric matrices, then (i) AB - B – Class 12 Mathematics

NCERT Exemplar · Q74

A

(i)

AB - BA

is a _________.

(ii)

BA - 2AB

is a _________.

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For symmetric matrices $A$ and $B$ , the commutator $AB - BA$ is always skew-symmetric, while $BA - 2AB$ is neither symmetric nor skew-symmetric in general — it is simply an arbitrary matrix.

Why This Approach Works

The key idea is simple: a symmetric matrix equals its own transpose, and a skew-symmetric matrix equals the negative of its transpose. When you multiply symmetric matrices, the transpose of the product reverses the order: $(AB)^T = B^T A^T = BA$ (since $A^T = A$ , $B^T = B$ ). So the transpose of $AB$ is $BA$ , not $AB$ itself. This reversal is the engine behind the entire problem.

We are asked to classify two expressions: $AB - BA$ and $BA - 2AB$ . The first is a classic commutator — its behaviour under transpose is well-known. The second is a linear combination that does not have a standard name. We will check each by taking its transpose and comparing with the original.

Watch out

A common mistake is to assume $AB$ is symmetric just because $A$ and $B$ are. That is false unless $A$ and $B$ commute. Always remember: $(AB)^T = B^T A^T$ , not $A^T B^T$ .

Step-by-Step Solution

1. Recall the definitions

A matrix $M$ is symmetric if $M^T = M$ .

A matrix $M$ is skew-symmetric if $M^T = -M$ .

Given: $A^T = A$ and $B^T = B$ .

2. Transpose of a product

For any matrices of compatible sizes:

(AB)^T = B^T A^T

Since $A$ and $B$ are symmetric, this becomes:

(AB)^T = BA

Tip

This is the only fact you need. The transpose flips the order, and symmetry removes the transposes on the individual matrices.

3. Analyse $AB - BA$

Let $C = AB - BA$ . Take its transpose:

C^T = (AB - BA)^T = (AB)^T - (BA)^T = BA - AB

But $BA - AB = -(AB - BA) = -C$ . So:

C^T = -C

This is exactly the definition of a skew-symmetric matrix.

A

Thus, the first blank is skew-symmetric.

4. Analyse $BA - 2AB$

Let $D = BA - 2AB$ . Take its transpose:

D^T = (BA - 2AB)^T = (BA)^T - 2(AB)^T = AB - 2BA

Now compare $D^T$ with $D$ :

For $D$ to be symmetric, we would need $D^T = D$ , i.e. $AB - 2BA = BA - 2AB$ .
For $D$ to be skew-symmetric, we would need $D^T = -D$ , i.e. $AB - 2BA = -(BA - 2AB) = 2AB - BA$ . …

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Q.If $A$ and $B$ are symmetric matrices, then

Concept understanding — Symmetric And Skew Symmetric Matrices

Symmetric and Skew-Symmetric Matrices

Symmetric matrix

Skew-symmetric matrix

Key facts

Why This Approach Works

Step-by-Step Solution

1. Recall the definitions

2. Transpose of a product

3. Analyse $AB - BA$

4. Analyse $BA - 2AB$

Method: Classify combinations of symmetric matrices

Steps

Common Mistakes

Q.If AAA and BBB are symmetric matrices, then

Why This Approach Works

Step-by-Step Solution

1. Recall the definitions

2. Transpose of a product

3. Analyse AB−BAAB - BAAB−BA

4. Analyse BA−2ABBA - 2ABBA−2AB

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Q.If $A$ and $B$ are symmetric matrices, then

3. Analyse $AB - BA$

4. Analyse $BA - 2AB$

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