- A$3$
- B$\frac{2}{3}$
- ✓$\frac{1}{3}$
- D$\frac{1}{{\sqrt {10} }}$
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$E=\left[\begin{array}{ccc}1 & 2 & 3 \\ 2 & 3 & 4 \\ 8 & 13 & 18\end{array}\right], P=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]$ and $F=\left[\begin{array}{ccc}1 & 3 & 2 \\ 8 & 18 & 13 \\ 2 & 4 & 3\end{array}\right]$
If $Q$ is a nonsingular matrix of order $3 \times 3$, then which of the following statements is (are) $TRUE$?
$(A)$F $=P E P$ and $P^2=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$
$(B)$ $\left| EQ + PFQ ^{-1}\right|=| EQ |+\left| PFQ ^{-1}\right|$
$(C)$ $\left|( EF )^3\right|>| EF |^2$
$(D)$ Sum of the diagonal entries of $P ^{-1} EP + F$ is equal to the sum of diagonal entries of $E + P ^{-1} FP$