Let A, B, C be 3 × 3 matrices such that A is symmetric and B and C are skew-symmetric. Consider the

Q: Wait, why does $(B^{26})^T$ become $B^{26}$? Shouldn't it be $-B^{26}$ because $B$ is skew-symmetric?

Great question! Remember that $(B^n)^T = (B^T)^n$. Since $B$ is skew-symmetric, $B^T = -B$. So, $(B^{26})^T = (-B)^{26}$. Because $26$ is an even power, the negative sign is absorbed: $(-1)^{26} = 1$. Thus, it becomes $B^{26}$. If the power were odd, the negative would survive!

Q: Can we just say $AB = BA$ because they are matrices? Why do we have to keep the order?

Never assume commutativity! In the world of matrices, $AB$ is rarely equal to $BA$. That is exactly why the Reversal Law $(AB)^T = B^T A^T$ is so vital—it forces us to respect the order of operations. If you ignore the order, you lose the physics of the problem.

Q: Is there a shortcut? Do I really have to write out the transpose for every single term?

In JEE Advanced, shortcuts are dangerous. Writing out the transpose step-by-step is the only way to avoid the 'sign trap.' The beauty of this problem lies in the symmetry of the steps. Once you master the process, you won't need to write every line, but for now, trust the process!

Question

Let $A, B, C$ be $3 	imes 3$ matrices such that $A$ is symmetric and $B$ and $C$ are skew-symmetric. Consider the statements (S1) $A^{13} B^{26} - B^{26} A^{13}$ is symmetric (S2) $A^{26} C^{13} - C^{13} A^{26}$ is symmetric. Then,

Options:

(A) Only S2 is true

(B) Only S1 is true

(C) Both S1 and S2 are false

(D) Both S1 and S2 are true

Sigma Percentile · Accepted Answer

### Analyzing the Setup

My dear students, welcome to a fascinating exploration of matrix properties. Today, we are not just solving a problem; we are uncovering the hidden geometry of linear algebra.

We are given three matrices: $A$, which is symmetric, and $B$ and $C$, which are skew-symmetric. Before we even touch the statements, let us ground ourselves in the definitions.

A symmetric matrix is one that is invariant under transposition, meaning $A^T = A$. A skew-symmetric matrix, however, is a bit more rebellious; it negates itself under transposition, so $B^T = -B$ and $C^T = -C$.

### The Dance of the Transpose

To analyze the statements, we need two powerful tools. First, the Reversal Law: $(XY)^T = Y^T X^T$. Imagine two dancers, $X$ and $Y$, holding hands. When they transpose, they don't just flip; they swap places!

Second, the Power Property: $(A^n)^T = (A^T)^n$. This tells us that the order of exponentiation and transposition does not matter. These two tools are our compass as we navigate the statements.

### Analyzing Statement $S_1$

Let us define our first expression as $M = A^{13} B^{26} - B^{26} A^{13}$. To check if $M$ is symmetric, we must compute $M^T$.

Applying the transpose to the difference, we get:
$$M^T = (A^{13} B^{26})^T - (B^{26} A^{13})^T$$

Now, we apply the Reversal Law:
$$M^T = (B^{26})^T (A^{13})^T - (A^{13})^T (B^{26})^T$$

We know $A^T = A$, so $(A^{13})^T = A^{13}$. For $B$, we have $(B^{26})^T = (B^T)^{26} = (-B)^{26}$. Since $26$ is an even power, the negative sign vanishes, leaving us with $B^{26}$.

Substituting these back, we find:
$$M^T = B^{26} A^{13} - A^{13} B^{26}$$

If we factor out a negative sign, we see $M^T = -(A^{13} B^{26} - B^{26} A^{13}) = -M$. Thus, $M$ is skew-symmetric, and **$S_1$ is false**.

### Analyzing Statement $S_2$

Now, let us turn our attention to $N = A^{26} C^{13} - C^{13} A^{26}$. We follow the same path:
$$N^T = (A^{26} C^{13})^T - (C^{13} A^{26})^T$$

Applying the Reversal Law, we get:
$$N^T = (C^{13})^T (A^{26})^T - (A^{26})^T (C^{13})^T$$

Again, $A^T = A$, so $(A^{26})^T = A^{26}$. But look at $C$: $(C^{13})^T = (C^T)^{13} = (-C)^{13}$. Since $13$ is an odd power, the negative sign survives, resulting in $-C^{13}$.

Substituting these into our equation:
$$N^T = (-C^{13}) A^{26} - A^{26} (-C^{13})$$

This simplifies to $N^T = -C^{13} A^{26} + A^{26} C^{13}$, which is exactly $N$. Since $N^T = N$, **statement $S_2$ is true**.

### The Final Revelation

We have walked through the logic, respected the order of operations, and carefully handled the even and odd powers. We found that $S_1$ is skew-symmetric and $S_2$ is symmetric.

Therefore, **only $S_2$ is true**. This problem teaches us that in mathematics, as in life, the small details—like whether a power is even or odd—can change the entire outcome.

Animated Solution for Mathematics - Matrices and Determinants: Let $A, B, C$ be $3 \times 3$ matrices such that $A$ is symmetric and $B$ and $C$ are skew-symmetric. Consider the statements (S1) $A^{13} B^{26} - B^{26} A^{13}$ is symmetric (S2) $A^{26} C^{13} - C^{13} A^{26}$ is symmetric. Then,

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Visualized Solution

Given Conditions

The Reversal Law of Transposes

Analyzing Statement $S_{1}$ : Setup

$S_{1}$ : Distributing the Transpose

$S_{1}$ : Applying the Reversal Law

$S_{1}$ : Handling Even and Odd Powers

$S_{1}$ : Final Substitution and Result

Analyzing Statement $S_{2}$ : Setup

$S_{2}$ : Applying the Reversal Law

$S_{2}$ : Handling the Odd Power of C

$S_{2}$ : Final Substitution and Result

Final Conclusion

The Sigma Insight: Types of Matrices

Analyzing the Setup

The Dance of the Transpose

Analyzing Statement $S_{1}$

Analyzing Statement $S_{2}$

The Final Revelation

Similar Questions

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.

Let $A$ and $B$ be any two $3 \times 3$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

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If A is a symmetric matrix and B is a skew-symmetric matrix such that $A + B = (25 3 - 1)$ , then AB is equal to :

Comprehension Passage

The number of $A$ in $T_{p}$ such that $A$ is either symmetric or skew-symmetric or both, and $det (A)$ divisible by $p$ is

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Let $A$ be a symmetric matrix such that $∣ A ∣ = 2$ and $[23 1 \frac{3}{2}] A = [1 α 2 β]$ . If the sum of the diagonal elements of $A$ is $s$ , then $\frac{β s}{α ^{2}}$ is equal to ______.

Let $A = [2 a 30], a \in R$ be written as $P + Q$ where $P$ is a symmetric matrix and $Q$ is a skew symmetric matrix. If $det (Q) = 9$ , then the modulus of the sum of all possible values of determinant of $P$ is equal to :

Let $A = 00 - 1 0 - 1 0 - 1 00$ . The only correct statement about the matrix $A$ is

The number of symmetric matrices of order 3, with all the entries from the set ${0, 1, 2, 3, 4, 5, 6, 7, 8, 9}$ is

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Visualized Solution

Given Conditions

The Reversal Law of Transposes

Analyzing Statement .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S1​: Setup

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S1​: Distributing the Transpose

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S1​: Applying the Reversal Law

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S1​: Handling Even and Odd Powers

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S1​: Final Substitution and Result

Analyzing Statement .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S2​: Setup

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S2​: Applying the Reversal Law

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S2​: Handling the Odd Power of C

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S2​: Final Substitution and Result