Key Concepts and Formulas

1. Symmetric Matrix: A square matrix $P$ is called symmetric if its transpose is equal to itself, i.e., $P^T = P$.
2. Skew-Symmetric Matrix: A square matrix $P$ is called skew-symmetric if its transpose is equal to its negative, i.e., $P^T = -P$.
3. Properties of Transpose: For any matrices $P$ and $Q$ (of compatible dimensions) and scalar $k$:
   • $(P \pm Q)^T = P^T \pm Q^T$ (The transpose of a sum or difference is the sum or difference of the transposes).
   • $(PQ)^T = Q^T P^T$ (The transpose of a product is the product of the transposes in reverse order).
   • $(kP)^T = kP^T$ (The transpose of a scalar multiple is the scalar multiple of the transpose).
   • $(P^k)^T = (P^T)^k$ for any positive integer $k$ (The transpose of a matrix raised to a power is the transpose of the matrix raised to that same power).

Problem Setup

We are given two $3 \times 3$ matrices:

• $A$ is a symmetric matrix, which implies $A^T = A$.
• $B$ is a skew-symmetric matrix, which implies $B^T = -B$.

Our objective is to determine which of the given statements regarding combinations of $A$ and $B$ is NOT true. We will evaluate each option by calculating the transpose of the given matrix expression and comparing it with the original expression or its negative.

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Step-by-Step Solution

Option (A): $A^4 - B^4$ is a symmetric matrix

Let $M = A^4 - B^4$. For $M$ to be symmetric, its transpose $M^T$ must be equal to $M$.

1. Calculate the transpose of $M$: $M^T = (A^4 - B^4)^T$ Reasoning: We apply the transpose operation to the entire matrix expression.

2. Apply the property $(P-Q)^T = P^T - Q^T$: $M^T = (A^4)^T - (B^4)^T$ Reasoning: The transpose of a difference of matrices is the difference of their transposes.

3. Apply the property $(P^k)^T = (P^T)^k$: $M^T = (A^T)^4 - (B^T)^4$ Reasoning: The transpose of a matrix raised to a power is equivalent to raising the transpose of the matrix to that same power.

4. Substitute the given properties of $A$ and $B$: Since $A$ is symmetric, $A^T = A$. Since $B$ is skew-symmetric, $B^T = -B$. $M^T = (A)^4 - (-B)^4$ Reasoning: We replace $A^T$ with $A$ and $B^T$ with $-B$ based on their definitions.

5. Simplify the terms: For any matrix $X$ and an even integer $k$, $(-X)^k = (-1)^k X^k = 1 \cdot X^k = X^k$. Thus, $(-B)^4 = B^4$. $M^T = A^4 - B^4$ Reasoning: We simplify the term $(-B)^4$ using the property of even powers.

6. Compare $M^T$ with $M$: We found $M^T = A^4 - B^4$, which is exactly equal to our original matrix $M$. Therefore, $M^T = M$.

Conclusion for Option (A): The matrix $A^4 - B^4$ is symmetric. This statement is TRUE.

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Option (B): $AB - BA$ is a symmetric matrix

Let $M = AB - BA$. For $M$ to be symmetric, its transpose $M^T$ must be equal to $M$.

1. Calculate the transpose of $M$: $M^T = (AB - BA)^T$ Reasoning: We apply the transpose operation to the entire expression.

2. Apply the property $(P-Q)^T = P^T - Q^T$: $M^T = (AB)^T - (BA)^T$ Reasoning: The transpose of a difference is the difference of the transposes.

3. Apply the property $(PQ)^T = Q^T P^T$: $M^T = B^T A^T - A^T B^T$ Reasoning: The transpose of a product of matrices requires reversing the order of multiplication of the transposes. This is a crucial step to remember.

4. Substitute the given properties of $A$ and $B$: Since $A^T = A$ and $B^T = -B$: $M^T = (-B)(A) - (A)(-B)$ Reasoning: We substitute $A^T$ with $A$ and $B^T$ with $-B$.

5. Simplify the terms: $M^T = -BA - (-AB)$ $M^T = -BA + AB$ $M^T = AB - BA$ Reasoning: We perform the matrix multiplications and simplify the signs.

6. Compare $M^T$ with $M$: We found $M^T = AB - BA$, which is exactly equal to our original matrix $M$. Therefore, $M^T = M$.

Conclusion for Option (B): The matrix $AB - BA$ is symmetric. This statement is TRUE.

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Option (C): $B^5 - A^5$ is a skew-symmetric matrix

Let $M = B^5 - A^5$. For $M$ to be skew-symmetric, its transpose $M^T$ must be equal to $-M$.

1. Calculate the transpose of $M$: $M^T = (B^5 - A^5)^T$ Reasoning: We apply the transpose operation to the entire expression.

2. Apply the property $(P-Q)^T = P^T - Q^T$: $M^T = (B^5)^T - (A^5)^T$ Reasoning: The transpose of a difference is the difference of the transposes.

3. Apply the property $(P^k)^T = (P^T)^k$: $M^T = (B^T)^5 - (A^T)^5$ Reasoning: The transpose of a matrix raised to a power is equivalent to raising the transpose of the matrix to that same power.

4. Substitute the given properties of $A$ and $B$: Since $A^T = A$ and $B^T = -B$: $M^T = (-B)^5 - (A)^5$ Reasoning: We substitute $A^T$ with $A$ and $B^T$ with $-B$.

5. Simplify the terms: For any matrix $X$ and an odd integer $k$, $(-X)^k = (-1)^k X^k = -1 \cdot X^k = -X^k$. Thus, $(-B)^5 = -B^5$. $M^T = -B^5 - A^5$ $M^T = -(B^5 + A^5)$ Reasoning: We simplify the term $(-B)^5$ using the property of odd powers.

6. Compare $M^T$ with $-M$: Our original matrix is $M = B^5 - A^5$. Therefore, $-M = -(B^5 - A^5) = -B^5 + A^5 = A^5 - B^5$. We found $M^T = -(B^5 + A^5) = -B^5 - A^5$. Clearly, $M^T \neq -M$ because $-B^5 - A^5$ is not generally equal to $A^5 - B^5$ (unless $A^5$ is the zero matrix).

Conclusion for Option (C): The matrix $B^5 - A^5$ is NOT a skew-symmetric matrix. This statement is FALSE. Since the question asks for the statement that is NOT true, this is our answer.

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Option (D): $AB + BA$ is a skew-symmetric matrix

Let $M = AB + BA$. For $M$ to be skew-symmetric, its transpose $M^T$ must be equal to $-M$.

1. Calculate the transpose of $M$: $M^T = (AB + BA)^T$ Reasoning: We apply the transpose operation to the entire expression.

2. Apply the property $(P+Q)^T = P^T + Q^T$: $M^T = (AB)^T + (BA)^T$ Reasoning: The transpose of a sum is the sum of the transposes.

3. Apply the property $(PQ)^T = Q^T P^T$: $M^T = B^T A^T + A^T B^T$ Reasoning: The transpose of a product requires reversing the order of multiplication of the transposes.

4. Substitute the given properties of $A$ and $B$: Since $A^T = A$ and $B^T = -B$: $M^T = (-B)(A) + (A)(-B)$ Reasoning: We substitute $A^T$ with $A$ and $B^T$ with $-B$.

5. Simplify the terms: $M^T = -BA + (-AB)$ $M^T = -BA - AB$ $M^T = -(AB + BA)$ Reasoning: We perform the matrix multiplications and factor out the negative sign.

6. Compare $M^T$ with $-M$: We found $M^T = -(AB + BA)$, which is exactly equal to $-M$. Therefore, $M^T = -M$.

Conclusion for Option (D): The matrix $AB + BA$ is skew-symmetric. This statement is TRUE.

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Common Mistakes & Tips

• Transpose of a Product: Always remember that $(PQ)^T = Q^T P^T$. A common error is to write $(PQ)^T = P^T Q^T$.
• Powers of Skew-Symmetric Matrices: For a skew-symmetric matrix $B$ ($B^T = -B$):
  • An even power of $B$ (e.g., $B^2, B^4$) results in a symmetric matrix. This is because $(B^k)^T = (B^T)^k = (-B)^k = B^k$ for even $k$.
  • An odd power of $B$ (e.g., $B^3, B^5$) results in a skew-symmetric matrix. This is because $(B^k)^T = (B^T)^k = (-B)^k = -B^k$ for odd $k$.
• Systematic Approach: For problems involving properties of matrix transposes, consistently calculate the transpose of the given expression step-by-step, substitute the known properties of the matrices, simplify, and then compare with the definition of symmetric or skew-symmetric matrices.

Summary

We systematically checked each option:

• Statement (A) is TRUE ($A^4 - B^4$ is symmetric).
• Statement (B) is TRUE ($AB - BA$ is symmetric).
• Statement (C) is FALSE ($B^5 - A^5$ is not skew-symmetric).
• Statement (D) is TRUE ($AB + BA$ is skew-symmetric).

The question asks for the statement that is NOT true. Our analysis shows that option (C) is the only statement that is false.

The final answer is $\boxed{C}$