Question

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

• A. $F=PEP$ and $P^2=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
• B. $\left|EQ+PF{Q}^{-1}\right|=|EQ|+\left|PF{Q}^{-1}\right|$
• 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 $E+P^{-1}FP$

Answer 1

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

Let $A=\left[\begin{array}{ccc}{C}_1& C_2& C_3\end{array}\right],B=\left[\begin{array}{c}{R}_1\\ R_2\\ R_3\end{array}\right]$
$AP=\left[\begin{array}{ccc}{C}_1& C_3& C_2\end{array}\right]$ and $PB=\left[\begin{array}{c}{R}_1\\ R_3\\ R_2\end{array}\right]$
and $P^2=I$
$P(EP)=P\left[\begin{array}{ccc}1& 3& 2\\ 2& 4& 3\\ 8& 18& 13\end{array}\right]=\left[\begin{array}{ccc}1& 3& 2\\ 8& 18& 13\\ 2& 4& 3\end{array}\right]=F$
$|E|=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 8& 13& 18\end{array}\right]\left(R_3\to R_3-3R_2-2R_1\right)=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 0& 0& 0\end{array}\right]=0$
$\Rightarrow |F|=0$
$\left|EQ+PF{Q}^{-1}\right|=\left|EQ+PPEP{Q}^{-1}\right|=\left|EQ+EP{Q}^{-1}\right|=|E|\left|Q+P{Q}^{-1}\right|=0$
$|EQ|=|E||Q|=0,\left|PF{Q}^{-1}\right|=|P||F|\left|Q^{-1}\right|=0$
(D) $P^{-1}EP+F=PEP+F=2F($ as $P^{-1}=P)$
$E+P^{-1}FP=E+P^{-1}PEPP=2E($ trace $(E)=$ trace $(F))$