Let A and B be any two $$3 \times 3$$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

We are given that A is a $$3 \times 3$$ symmetric matrix (so $$A^T = A$$) and B is a $$3 \times 3$$ skew-symmetric matrix (so $$B^T = -B$$). We need to find which statement is NOT true.

Option 1: $$A^4 - B^4$$ is symmetric. We check: $$(A^4 - B^4)^T = (A^T)^4 - (B^T)^4 = A^4 - (-B)^4 = A^4 - B^4$$. So $$A^4 - B^4$$ is symmetric. This is TRUE.

Option 2: $$AB - BA$$ is symmetric. We check: $$(AB - BA)^T = (AB)^T - (BA)^T = B^T A^T - A^T B^T = (-B)(A) - (A)(-B) = -BA + AB = AB - BA$$. So $$AB - BA$$ is symmetric. This is TRUE.

Option 3: $$B^5 - A^5$$ is skew-symmetric. We check: $$(B^5 - A^5)^T = (B^T)^5 - (A^T)^5 = (-B)^5 - A^5 = -B^5 - A^5 = -(B^5 + A^5)$$. For this to be skew-symmetric, we need $$(B^5 - A^5)^T = -(B^5 - A^5) = -B^5 + A^5$$. But we got $$-B^5 - A^5$$. These are equal only if $$A^5 = -A^5$$, i.e., $$A^5 = 0$$, which is not generally true. So this is NOT TRUE.

Option 4: $$AB + BA$$ is skew-symmetric. We check: $$(AB + BA)^T = B^T A^T + A^T B^T = (-B)(A) + (A)(-B) = -BA - AB = -(AB + BA)$$. So $$AB + BA$$ is skew-symmetric. This is TRUE.

Hence, the correct answer is Option 3.