A) $\mathbf{F} = \mathbf{P}\mathbf{E}\mathbf{P}$ and $\mathbf{P}^2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
B) $|\mathbf{E}\mathbf{Q} + \mathbf{P}\mathbf{F}\mathbf{Q}^{-1}| = |\mathbf{E}\mathbf{Q}| + |\mathbf{P}\mathbf{F}\mathbf{Q}^{-1}|$
C) $|(\mathbf{E}\mathbf{F})^3| > |\mathbf{E}\mathbf{F}|^2$
D) Sum of the diagonal entries of $\mathbf{P}^{-1}\mathbf{E}\mathbf{P} + \mathbf{F}$ is equal to the sum of diagonal entries of $\mathbf{E} + \mathbf{P}^{-1}\mathbf{F}\mathbf{P}$
Step-by-Step Solution
Key Concept: A nonsingular matrix has an inverse if and only if its determinant is nonzero.
$$mathbf{Q}$ is a nonsingular matrix, so it has an inverse. Since $mathbf{Q}$ is $3 \times 3$, we can use the property that a matrix has an inverse if and only if its determinant is nonzero. Therefore, the determinant of $mathbf{Q}$ must be nonzero. This implies that the statement that $mathbf{Q}$ has an inverse is TRUE. Additionally, since $mathbf{Q}$ is nonsingular, it is also invertible, so the statement that $mathbf{Q}$ is invertible is TRUE. However, the statement that $mathbf{Q}$ is singular is FALSE, as it contradicts the fact that $mathbf{Q}$ is nonsingular. Therefore, the correct answer is that the statements that $mathbf{Q}$ has an inverse and $mathbf{Q}$ is invertible are TRUE.
Correct Answer: B, C, D