Key Concepts and Definitions

Before diving into the problem, let's recall the fundamental definitions and properties related to symmetric and skew-symmetric matrices:

1. Symmetric Matrix: A square matrix $X$ is called symmetric if its transpose is equal to itself, i.e., $X^T = X$.
2. Skew-Symmetric Matrix: A square matrix $X$ is called skew-symmetric if its transpose is equal to its negative, i.e., $X^T = -X$.

Important Properties of Transpose:

For any matrices $X$ and $Y$ of appropriate dimensions and any scalar $k$:

• $(X+Y)^T = X^T + Y^T$
• $(XY)^T = Y^T X^T$ (Note the reversal of order)
• $(kX)^T = kX^T$
• $(X^n)^T = (X^T)^n$ for any positive integer $n$.

Properties of Powers of Symmetric/Skew-Symmetric Matrices:

• If $X$ is symmetric ($X^T = X$), then $X^n$ is always symmetric for any positive integer $n$.
  • Proof: $(X^n)^T = (X^T)^n = X^n$.
• If $X$ is skew-symmetric ($X^T = -X$):
  • $X^n$ is symmetric if $n$ is an even positive integer.
    • Proof: $(X^n)^T = (X^T)^n = (-X)^n = (-1)^n X^n$. If $n$ is even, $(-1)^n = 1$, so $(X^n)^T = X^n$.
  • $X^n$ is skew-symmetric if $n$ is an odd positive integer.
    • Proof: $(X^n)^T = (X^T)^n = (-X)^n = (-1)^n X^n$. If $n$ is odd, $(-1)^n = -1$, so $(X^n)^T = -X^n$.

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Given Information

We are given three $3 \times 3$ matrices A, B, and C with the following properties:

• A is symmetric, which means $A^T = A$.
• B is skew-symmetric, which means $B^T = -B$.
• C is skew-symmetric, which means $C^T = -C$.

We need to evaluate the truthfulness of two statements, (S1) and (S2).

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Analysis of Statement (S1)

Statement (S1) says that $P = A^{13} B^{26} - B^{26} A^{13}$ is symmetric. To check this, we need to find the transpose of $P$ and see if it equals $P$ or $-P$.

Step 1: Determine the nature of $A^{13}$ and $B^{26}$.

• For $A^{13}$: Since A is symmetric ($A^T=A$) and $13$ is an integer power, $A^{13}$ will also be symmetric.
  • Let's verify: $(A^{13})^T = (A^T)^{13} = A^{13}$.
• For $B^{26}$: Since B is skew-symmetric ($B^T=-B$) and $26$ is an even power, $B^{26}$ will be symmetric.
  • Let's verify: $(B^{26})^T = (B^T)^{26} = (-B)^{26} = (-1)^{26} B^{26} = 1 \cdot B^{26} = B^{26}$.

So, both $A^{13}$ and $B^{26}$ are symmetric matrices. Let's denote $X = A^{13}$ and $Y = B^{26}$. We know $X^T = X$ and $Y^T = Y$.

Step 2: Calculate the transpose of $P$.

We have $P = XY - YX$. Now we find $P^T$: $P^T = (XY - YX)^T$ Using the property $(M-N)^T = M^T - N^T$: $P^T = (XY)^T - (YX)^T$ Using the property $(MN)^T = N^T M^T$ for matrix products: $P^T = Y^T X^T - X^T Y^T$ Now, substitute $X^T = X$ and $Y^T = Y$ (since both $A^{13}$ and $B^{26}$ are symmetric): $P^T = YX - XY$ This can be rewritten as: $P^T = -(XY - YX)$ Since $P = XY - YX$, we have: $P^T = -P$

Step 3: Conclusion for (S1).

Since $P^T = -P$, the matrix $P$ is skew-symmetric. Therefore, statement (S1) is false.

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Analysis of Statement (S2)

Statement (S2) says that $Q = A^{26} C^{13} - C^{13} A^{26}$ is symmetric. Again, we find the transpose of $Q$.

Step 1: Determine the nature of $A^{26}$ and $C^{13}$.

• For $A^{26}$: Since A is symmetric ($A^T=A$) and $26$ is an integer power, $A^{26}$ will also be symmetric.
  • Let's verify: $(A^{26})^T = (A^T)^{26} = A^{26}$.
• For $C^{13}$: Since C is skew-symmetric ($C^T=-C$) and $13$ is an odd power, $C^{13}$ will be skew-symmetric.
  • Let's verify: $(C^{13})^T = (C^T)^{13} = (-C)^{13} = (-1)^{13} C^{13} = -C^{13}$.

So, $A^{26}$ is symmetric, and $C^{13}$ is skew-symmetric. Let's denote $M = A^{26}$ and $N = C^{13}$. We know $M^T = M$ and $N^T = -N$.

Step 2: Calculate the transpose of $Q$.

We have $Q = MN - NM$. Now we find $Q^T$: $Q^T = (MN - NM)^T$ Using the property $(X-Y)^T = X^T - Y^T$: $Q^T = (MN)^T - (NM)^T$ Using the property $(XY)^T = Y^T X^T$ for matrix products: $Q^T = N^T M^T - M^T N^T$ Now, substitute $M^T = M$ and $N^T = -N$: $Q^T = (-N)M - M(-N)$ $Q^T = -NM + MN$ This can be rewritten by rearranging the terms: $Q^T = MN - NM$ Since $Q = MN - NM$, we have: $Q^T = Q$

Step 3: Conclusion for (S2).

Since $Q^T = Q$, the matrix $Q$ is symmetric. Therefore, statement (S2) is true.

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Final Conclusion

Based on our analysis:

• Statement (S1) is false.
• Statement (S2) is true.

Comparing this with the given options, option (A) "Only S2 is true" is the correct answer.

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Tips and Common Mistakes to Avoid

1. Don't forget the order reversal for $(XY)^T = Y^T X^T$: This is a very common mistake. Always remember to reverse the order of multiplication when taking the transpose of a product.
2. Be careful with signs when dealing with odd/even powers of skew-symmetric matrices:
   • $(-X)^{\text{even power}} = X^{\text{even power}}$
   • $(-X)^{\text{odd power}} = -X^{\text{odd power}}$ This distinction is critical for determining the nature of powers of skew-symmetric matrices.
3. Understand the definitions clearly: A matrix $X$ is symmetric if $X^T = X$, and skew-symmetric if $X^T = -X$. Always refer back to these definitions