Let X and Y be two arbitrary, 3 x 3, non-zero, skew-symmetric matrices and Z be an arbitrary 3 x 3,

Q: Why does the order flip in $(AB)^T = B^T A^T$?

Think of matrix multiplication as a sequence of transformations. When you transpose the entire operation, you are essentially reversing the sequence of events. If you perform operation A then B, the transpose reverses the order of the operations themselves. It is not just a property; it is the geometric necessity of how matrices interact.

Q: Can I just assume $X$ and $Y$ are diagonal matrices to test this?

While testing with specific matrices is a great way to build intuition, it is dangerous in a JEE Advanced context. A property might hold for diagonal matrices but fail for general matrices. Always rely on the algebraic definitions ($X^T = -X$) to ensure your proof is rigorous and universal.

Q: Why does the negative sign disappear for even powers but stay for odd powers?

This is the heart of the problem! When you have $(-X)^n$, you are essentially multiplying $(-1)$ by itself $n$ times. If $n$ is even, you have an even number of negative signs, which cancel out to become positive. If $n$ is odd, one negative sign remains. It is the same logic as $(-1)^2 = 1$ and $(-1)^3 = -1$.

Question

Let $X$ and $Y$ be two arbitrary, $3 	imes 3$, non-zero, skew-symmetric matrices and $Z$ be an arbitrary $3 	imes 3$, non zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?

Options:

(A) $Y^3 Z^4 - Z^4 Y^3$

(B) $X^{44} + Y^{44}$

(C) $X^4 Z^3 - Z^3 X^4$

(D) $X^{23} + Y^{23}$

Sigma Percentile · Accepted Answer

### The Dance of Symmetry: Mastering Matrix Transposes

Welcome, future engineers. Today, we are not just solving a problem; we are exploring the elegant, rhythmic dance of matrix algebra.

When you look at a problem involving skew-symmetric and symmetric matrices, it is easy to feel overwhelmed by the powers and the notation. But I want you to take a deep breath.

Matrix algebra is not about memorizing rules; it is about understanding the symmetry of the universe. Let us break this down, step by step, and uncover the hidden patterns within these matrices.

### Phase 1: The Rules of the Game

Before we dive into the options, we must define our players. We are given two skew-symmetric matrices, $X$ and $Y$, and one symmetric matrix, $Z$.

Mathematically, this gives us our fundamental constraints: 
$$X^T = -X, \quad Y^T = -Y, \quad Z^T = Z$$

Our goal is to identify which of the given options results in a skew-symmetric matrix. To do this, we must test each candidate matrix $M$ by checking if $M^T = -M$.

Think of the transpose operator as a mirror. If the mirror image of the matrix is its negative, we have found our skew-symmetric treasure.

### Phase 2: The Reversal Law

To perform this investigation, we need our toolkit. The most vital tool in your arsenal is the Reversal Law: 
$$(AB)^T = B^T A^T$$

Why does the order flip? Imagine you are performing a sequence of actions. When you reverse the entire process, the last action becomes the first.

This is the geometric soul of the reversal law. We also rely on the power rule: 
$$(A^n)^T = (A^T)^n$$

With these two tools, we can dismantle any matrix expression, no matter how complex it looks.

### Phase 3: The Investigation

Let us test our options. Consider Option A: $M_1 = Y^3 Z^4 - Z^4 Y^3$.

When we apply the transpose, we get $M_1^T = (Y^3 Z^4)^T - (Z^4 Y^3)^T$. Applying the reversal law, this becomes:
$$M_1^T = (Z^4)^T (Y^3)^T - (Y^3)^T (Z^4)^T$$

Substituting our known properties, $Z^T = Z$ and $Y^T = -Y$, we get $Z^4 (-Y)^3 - (-Y)^3 Z^4$. Since $3$ is an odd power, $(-Y)^3 = -Y^3$.

This simplifies to $-Z^4 Y^3 + Y^3 Z^4$, which is exactly $M_1$. Since $M_1^T = M_1$, this matrix is symmetric.

Now, look at Option B: $M_2 = X^{44} + Y^{44}$. Applying the transpose, we get $M_2^T = (X^T)^{44} + (Y^T)^{44}$.

Substituting $X^T = -X$ and $Y^T = -Y$, we get $(-X)^{44} + (-Y)^{44}$. Here is the magic: $44$ is an even number.

Any negative value raised to an even power becomes positive. Thus, $(-X)^{44} = X^{44}$. We are left with $M_2^T = X^{44} + Y^{44} = M_2$.

Again, we have a symmetric matrix. The even power has effectively 'absorbed' the skew-symmetry.

### Phase 4: The Breakthrough

Now, let us look at Option C: $M_3 = X^4 Z^3 - Z^3 X^4$. Applying the transpose and the reversal law, we get:
$$M_3^T = (Z^3)^T (X^4)^T - (X^4)^T (Z^3)^T$$

Substituting $Z^T = Z$ and $X^T = -X$, we get $Z^3 (-X)^4 - (-X)^4 Z^3$. Since $4$ is even, $(-X)^4 = X^4$.

This simplifies to $Z^3 X^4 - X^4 Z^3$. If we factor out a negative sign, we get $-(X^4 Z^3 - Z^3 X^4) = -M_3$.

Success! $M_3^T = -M_3$. This is **skew-symmetric**.

Finally, Option D: $M_4 = X^{23} + Y^{23}$. Applying the transpose, we get $M_4^T = (X^T)^{23} + (Y^T)^{23}$.

Substituting $X^T = -X$ and $Y^T = -Y$, we get $(-X)^{23} + (-Y)^{23}$. Because $23$ is an odd number, the negative sign persists: $(-X)^{23} = -X^{23}$.

Thus, $M_4^T = -X^{23} - Y^{23} = -(X^{23} + Y^{23}) = -M_4$. This is also **skew-symmetric**.

### Conclusion: The Pattern of Parity

We have arrived at our destination. The key takeaway is the parity of the exponent.

If a matrix $A$ is skew-symmetric, $A^n$ is symmetric if $n$ is even, and skew-symmetric if $n$ is odd. This is the elegant pattern that governs the behavior of these matrices.

You have successfully navigated the complexity of matrix algebra by relying on the fundamental definitions. Keep this logic close to your heart, and no matrix problem will ever intimidate you again.

Animated Solution for Mathematics - Matrices and Determinants: Let $X$ and $Y$ be two arbitrary, $3 \times 3$ , non-zero, skew-symmetric matrices and $Z$ be an arbitrary $3 \times 3$ , non zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?

Select Answer:

Visualized Solution

Given Matrix Properties

Rules of Transpose

Option A: Setup

Option A: Execution

Option B: Setup

Option B: Execution

Option C: Setup

Option C: Execution

Option D: Setup

Option D: Execution

Key Takeaways

The Sigma Insight: Types of Matrices

The Dance of Symmetry

Phase 1

Phase 2

Phase 3

Phase 4

Conclusion

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