If the matrix A is both symmetric and skew-symmetric, then

A) A is a diagonal matrix

B) A is a zero matrix

C) A is a square matrix

D) None of these

To tackle the question of what happens when a matrix \( A \) is both symmetric and skew-symmetric, let's first clarify the definitions of these terms. A matrix is symmetric if it is equal to its transpose, meaning \( A = A^T \). On the other hand, a matrix is skew-symmetric if it is equal to the negative of its transpose, which means \( A = -A^T \). Now, if a matrix satisfies both conditions, we can derive some interesting conclusions.

Analyzing the Conditions

Given that \( A \) is both symmetric and skew-symmetric, we can set up the following equations:

• From symmetry: \( A = A^T \)
• From skew-symmetry: \( A = -A^T \)

Now, if we substitute the first equation into the second, we get:

A = -A

This implies that:

2A = 0

From this, we can conclude that:

A = 0

Conclusion About the Matrix

Thus, the only matrix that can satisfy both conditions is the zero matrix. This means that if a matrix is both symmetric and skew-symmetric, it must be the zero matrix. Therefore, the correct answer to the question is:

B) A is a zero matrix

Understanding the Implications

This result has interesting implications in linear algebra. The zero matrix is unique in that it has all its entries equal to zero, and it serves as the additive identity in matrix operations. This means that when you add any matrix to the zero matrix, the result is the original matrix itself. Moreover, it also highlights the relationship between symmetry and skew-symmetry, showing that they cannot coexist in any non-zero form.

In summary, when a matrix is both symmetric and skew-symmetric, it can only be the zero matrix, reinforcing the idea that certain properties in linear algebra can lead to very specific outcomes. If you have any further questions or need clarification on any part of this explanation, feel free to ask!