Let A and B be two symmetric matrices of order 3. Statement-1: A(BA) and (AB)A are symmetric matrice

Q: Wait, why does the order of matrices flip when we take the transpose? Is this just a random rule?

It is not random at all! Think about the definition of matrix multiplication. The element in the $i$-th row and $j$-th column of $AB$ is the dot product of the $i$-th row of $A$ and the $j$-th column of $B$. When you transpose, rows become columns and columns become rows. The reversal is a direct consequence of this geometric transformation. It is the fundamental way matrices 'dance' together.

Q: Can't we just assume $A$ and $B$ are diagonal matrices to check the statements? That would be much faster.

While testing with specific cases like diagonal matrices can give you a quick intuition, it is a dangerous habit for JEE Advanced. A statement must hold true for *all* symmetric matrices. If you only test diagonal matrices, you might miss a case where the property fails for non-diagonal symmetric matrices. Always rely on the general algebraic proof using the Reversal Law.

Q: Is Statement-2 the reason Statement-1 is true? They both involve symmetry and commutativity.

That is the classic trap! Statement-1 is true because of the inherent properties of symmetric matrices ($A^T=A, B^T=B$). Statement-2 is a conditional statement about when $AB$ itself is symmetric. They are logically independent. We proved Statement-1 without ever needing the condition $AB=BA$. Therefore, Statement-2 is not the explanation for Statement-1.

Question

Let $A$ and $B$ be two symmetric matrices of order 3. \ Statement-1: $A(BA)$ and $(AB)A$ are symmetric matrices. \ Statement-2: $AB$ is symmetric matrix if matrix multiplication of $A$ with $B$ is commutative.

Options:

(A) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(B) Statement-1 is true, Statement-2 is false.

(C) Statement-1 is false, Statement-2 is true.

(D) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

In the world of linear algebra, symmetric matrices are defined by the elegant property $A^T = A$. This means that if you flip the matrix across its main diagonal, you obtain the exact same matrix.

Today, we explore the logical dance of these matrices through the lens of two specific statements regarding symmetric matrices $A$ and $B$.

### The Reversal Law: The Heartbeat of Matrix Algebra

Before we dive into the problem, we must utilize the **Reversal Law of Transpose**. It states that for any product of matrices, the transpose of the product is the product of the transposes in reverse order.

Mathematically, this is expressed as:

$$ (XYZ)^T = Z^T Y^T X^T $$

Think of this as a reversal of order. It is the essential key that unlocks the structure of any matrix product.

### Proving Statement-1: The Symmetry of $A(BA)$

To test if $A(BA)$ is symmetric, we apply the 'mirror test' by taking the transpose of the entire expression and checking if it returns to its original form.

Starting with $(A(BA))^T$, we apply the Reversal Law:

$$ (A(BA))^T = (BA)^T A^T $$

Applying the Reversal Law again to the inner term $(BA)^T$, we get $A^T B^T$. Substituting this back into our equation yields:

$$ (A(BA))^T = A^T B^T A^T $$

Given the properties $A^T = A$ and $B^T = B$, the expression simplifies to:

$$ A^T B^T A^T = ABA $$

Since $ABA$ is identical to the original expression $A(BA)$, we have successfully proven that **$A(BA)$ is symmetric**.

### The Commutativity Trap: Analyzing Statement-2

Statement-2 claims that $AB$ is symmetric if and only if $AB = BA$. Let us test this by taking the transpose of $AB$:

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

Using the symmetric properties $A^T = A$ and $B^T = B$, this simplifies to $BA$. For $AB$ to be symmetric, we require $(AB)^T = AB$, which implies:

$$ BA = AB $$

Thus, **Statement-2 is true**. However, we must determine if Statement-2 explains Statement-1.

We proved Statement-1 using only the fundamental properties of $A$ and $B$ being symmetric. We never required the assumption that $A$ and $B$ commute.

Therefore, while both statements are true, **Statement-2 is not the correct explanation for Statement-1**.

### The Final Reflection

In this problem, we have seen how the Reversal Law acts as a bridge between the abstract definition of symmetry and the concrete reality of matrix products.

We learned that while commutativity is a powerful condition, it is not the universal requirement for symmetry in complex products. Always look for the fundamental properties first, apply the laws step-by-step, and let the algebra reveal the truth.

Animated Solution for Mathematics - Matrices and Determinants: Let $A$ and $B$ be two symmetric matrices of order 3. \\ Statement-1: $A (B A)$ and $(A B) A$ are symmetric matrices. \\ Statement-2: $A B$ is symmetric matrix if matrix multiplication of $A$ with $B$ is commutative.

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Visualized Solution

Defining Symmetric Matrices

The Reversal Law of Transpose

Testing Symmetry of $A (B A)$

Applying the Reversal Law

Expanding the Inner Transpose

Substituting Given Properties

Testing Symmetry of $(A B) A$

Analyzing Statement-2

The Commutative Condition

Final Conclusion

The Sigma Insight: Types of Matrices

Analyzing the Setup

The Reversal Law

Proving Statement-1

The Commutativity Trap

The Final Reflection

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