Question

For any $3 \times 3$ matrix $M$, let $| M |$ denote the determinant of $M$. Let
$E=\begin{bmatrix}1 & 2 & 3 \\\2 & 3 & 4 \\\8 & 13 & 18\end{bmatrix}, P=\begin{bmatrix} 1 & 0 & 0 \\\0 & 0 & 1 \\\0 & 1 & 0\end{bmatrix}$ and $F=\begin{bmatrix} 1 & 3 & 2 \\\ 8 & 18 & 13 \\\ 2 & 4 & 3 \end{bmatrix}$
If $Q$ is a non-singular matrix of order $3 \times 3$, then which of the following statements is(are) TRUE?

• A. $F = PEP$ and $P ^{2}=\begin{bmatrix}1 & 0 & 0 \\\ 0 & 1 & 0 \\\ 0 & 0 & 1\end{bmatrix}$
• 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 $P^{-1} E P+F$ is equal to the sum of diagonal entries of $E+P^{-1} F P$

Answer 1

Answer: (A), (B), (D)

$F = PEP$ and $P ^{2}=\begin{bmatrix}1 & 0 & 0 \\\ 0 & 1 & 0 \\\ 0 & 0 & 1\end{bmatrix}$

Answer 2

The correct options are:
(A) $F = PEP$ and $P ^{2}=\begin{bmatrix}1 & 0 & 0 \\\ 0 & 1 & 0 \\\ 0 & 0 & 1\end{bmatrix}$
(B) $\left| EQ + PFQ ^{-1}\right|=| EQ |+\left| PFQ ^{-1}\right|$
(D) Sum of the diagonal entries of $P^{-1} E P+F$ is equal to the sum of diagonal entries of $E+P^{-1} F P$