Question

P and Q are square matrices. Consider the following:
X: $(P^{-1})^{-1} = P$
Y: Symmetric if $Q = -Q^T$
The correct choice is:

• A. X is TRUE; Y is FALSE
• B. X is FALSE; Y is TRUE
• C. Both X and Y are TRUE
• D. Both X and Y are FALSE

Hint:

Remember: $(A^{-1})^{-1} = A$. A matrix $Q$ is symmetric if $Q = Q^T$; skew-symmetric if $Q = -Q^T$.

Answer 1

The Correct Option is A

Step 1: Check statement X.
We know that the inverse of the inverse of a matrix is the matrix itself: \[ (P^{-1})^{-1} = P \] So X is TRUE.
Step 2: Check statement Y.
If $Q = -Q^T$, then $Q$ is skew-symmetric, not symmetric. So Y is FALSE. Final Answer: \[ \boxed{\text{(A) X is TRUE; Y is FALSE}} \]