Linear Algebra Proofs (Diction Understanding) for high school student

Question

I am doing a practice homework set where the question asserts if something is necessarily ________?

Does this imply that it is always true for all cases?

For example Given that X is a symmetric nxn matrix, and Y is a nxn skew-symmetric matrix, which of the following is NECESSARILY true.

1. XYX is symmetric
2. XYX is skew-symmetric

3. XY^2X is symmetric

4. XY^2X is skew-symmetric.

Going by the definition of necessarily being undeniably true, then I can conclude that 2 and 4 are accurate assumptions. However, are 1 and 4 necessarily skew-symmetric if I can list an example proving their validity?

Answer 1

1. $(XYX)^T=X^TY^TX^T=-XYX$. So, the statement "$XYX$ is symmetric" is false.

2. By the result above, we conclude that the statement "$XYX$ is skew-symmetric" is true.

3. $(XY^2X)^T=(XYYX)^T=X^TY^TY^TX^T=XYYX=XY^2X$. So, the statement "$XY^2X$ is symmetric" is true.

4. By the result above, we conclude that the statement "$XY^2X$ is skew-symmetric" is false.