This question asks us to identify the statement that is NOT true regarding skew-symmetric matrices A and B.

Let's recall the definition and properties of a skew-symmetric matrix. A matrix $M$ is skew-symmetric if its transpose $M^T$ is equal to its negative, i.e., $M^T = -M$.

Key Properties of Skew-Symmetric Matrices

• If $M$ is skew-symmetric, then $M^T = -M$.
• If $M$ is skew-symmetric, then $M^n$ is symmetric when $n$ is an even positive integer, because $(M^n)^T = (M^T)^n = (-M)^n = M^n$.
• If $M$ is skew-symmetric, then $M^n$ is skew-symmetric when $n$ is an odd positive integer, because $(M^n)^T = (M^T)^n = (-M)^n = -M^n$.
• If $M$ and $N$ are skew-symmetric matrices of the same dimension, their sum ($M+N$) and difference ($M-N$) are also skew-symmetric.
  • $(M+N)^T = M^T + N^T = (-M) + (-N) = -(M+N)$
  • $(M-N)^T = M^T - N^T = (-M) - (-N) = -M + N = -(M-N)$

Analysis of Options

Option 1: $A^3 + B^5$ is skew-symmetric

Given that A and B are skew-symmetric matrices:

• Since the power 3 is odd, $A^3$ is skew-symmetric. $(A^3)^T = (A^T)^3 = (-A)^3 = -A^3$.
• Since the power 5 is odd, $B^5$ is skew-symmetric. $(B^5)^T = (B^T)^5 = (-B)^5 = -B^5$.
• The sum of two skew-symmetric matrices ($A^3$ and $B^5$) is also skew-symmetric. $(A^3 + B^5)^T = (A^3)^T + (B^5)^T = (-A^3) + (-B^5) = -(A^3 + B^5)$.

Therefore, the statement "$A^3 + B^5$ is skew-symmetric" is TRUE.

Option 2: $A^{19}$ is skew-symmetric

Given that A is a skew-symmetric matrix:

• The power 19 is odd.
• Therefore, $A^{19}$ is skew-symmetric. $(A^{19})^T = (A^T)^{19} = (-A)^{19} = -A^{19}$.

Therefore, the statement "$A^{19}$ is skew-symmetric" is TRUE.

Option 3: $B^{14}$ is symmetric

Given that B is a skew-symmetric matrix:

• The power 14 is even.
• Therefore, $B^{14}$ is symmetric. $(B^{14})^T = (B^T)^{14} = (-B)^{14} = B^{14}$.

Therefore, the statement "$B^{14}$ is symmetric" is TRUE.

Option 4: $A^4 + B^5$ is symmetric

Given that A and B are skew-symmetric matrices:

• Since the power 4 is even, $A^4$ is symmetric. $(A^4)^T = (A^T)^4 = (-A)^4 = A^4$.
• Since the power 5 is odd, $B^5$ is skew-symmetric. $(B^5)^T = (B^T)^5 = (-B)^5 = -B^5$.
• Let $S = A^4 + B^5$. We check if $S^T = S$.
• $S^T = (A^4 + B^5)^T = (A^4)^T + (B^5)^T = A^4 + (-B^5) = A^4 - B^5$.
• For $S$ to be symmetric, we must have $S^T = S$, which means $A^4 - B^5 = A^4 + B^5$.
• This simplifies to $-B^5 = B^5$, or $2B^5 = 0$. This implies $B^5 = 0$.
• However, $B^5$ is not necessarily the zero matrix for any skew-symmetric matrix B. For example, if B is a non-zero skew-symmetric matrix, $B^5$ might not be zero.

Since the condition for $A^4 + B^5$ being symmetric ($B^5=0$) is not generally true, the statement "$A^4 + B^5$ is symmetric" is NOT necessarily true.

Conclusion

Based on the analysis, the statement that is NOT true is "$A^4 + B^5$ is symmetric".