If A and B are skew-symmetric matrices, then which one of the following is NOT true?

A matrix $M$ is skew-symmetric if:

$M^T = -M$

When you flip the matrix across its diagonal (transpose it), you get the negative of the original matrix.

Simple example: If $M = \begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}$, then $M^T = \begin{bmatrix} 0 & -2 \\ 2 & 0 \end{bmatrix} = -M$

Diagonal elements of skew-symmetric matrices are always zero.

---

When you raise a skew-symmetric matrix to a power:

Odd powers $(1, 3, 5, 7...)$ stay skew-symmetric

Even powers $(2, 4, 6, 8...)$ become symmetric

For a skew-symmetric matrix $M$ where $M^T = -M$:

$(M^2)^T = (M \cdot M)^T = M^T \cdot M^T = (-M)(-M) = M^2$

So $M^2$ is symmetric.

$(M^3)^T = (M^2 \cdot M)^T = M^T \cdot (M^2)^T = (-M) \cdot M^2 = -M^3$

So $M^3$ is skew-symmetric.

---

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

$A^3$ is skew-symmetric (odd power)

$B^5$ is skew-symmetric (odd power)

Skew-symmetric $+$ Skew-symmetric $=$ Skew-symmetric

This is TRUE.

---

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

$A^{19}$ has odd power, so it is skew-symmetric.

This is TRUE.

---

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

$B^{14}$ has even power, so it is symmetric.

This is TRUE.

---

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

$A^4$ is symmetric (even power)

$B^5$ is skew-symmetric (odd power)

Checking using transpose:

$(A^4 + B^5)^T = (A^4)^T + (B^5)^T = A^4 + (-B^5) = A^4 - B^5$

For the sum to be symmetric, we need $(A^4 + B^5)^T = A^4 + B^5$

But we got $A^4 - B^5 \neq A^4 + B^5$

This is FALSE.

---

Option 4 is NOT true.