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)

Let $A =\begin{bmatrix} C _{1} & C _{2} & C _{3}\end{bmatrix}, B =\begin{bmatrix} R _{1} \\ R _{2} \\ R _{3}\end{bmatrix}$
$AP =\begin{bmatrix} C _{1} & C _{3} & C _{2}\end{bmatrix}$ and $ PB =\begin{bmatrix} R _{1} \\ R _{3} \\ R _{2}\end{bmatrix}$
and $P ^{2}= I$
$P ( EP )= P \begin{bmatrix}1 & 3 & 2 \\ 2 & 4 & 3 \\ 8 & 18 & 13\end{bmatrix}=\begin{bmatrix}1 & 3 & 2 \\ 8 & 18 & 13 \\ 2 & 4 & 3\end{bmatrix}= F$
$| E |=\begin{bmatrix}1 & 2 & 3 \\ 2 & 3 & 4 \\ 8 & 13 & 18\end{bmatrix}\left( R _{3} \rightarrow R _{3}-3 R _{2}-2 R _{1}\right)=\begin{bmatrix}1 & 2 & 3 \\ 2 & 3 & 4 \\ 0 & 0 & 0\end{bmatrix}=0$
$\Rightarrow | F |=0$
$\left| EQ + PFQ ^{-1}\right|=\left| EQ + PPEPQ ^{-1}\right|=\left| EQ + EPQ ^{-1}\right|=| E |\left| Q + PQ ^{-1}\right|=0$
$| EQ |=| E || Q |=0,\left| PFQ ^{-1}\right|=| P || F |\left| Q ^{-1}\right|=0$
(D) $P ^{-1} EP + F = PEP + F =2 F \left(\right.$ as $\left.P ^{-1}= P \right)$
$E + P ^{-1} FP = E + P ^{-1} PEPP =2 E ($ trace $( E )=$ trace $( F ))$