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 nonsingular 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. $|EQ+PF{Q}^{-1}|=|EQ|+|PF{Q}^{-1}|$
• C. $|(EF{)}^3|>|EF{|}^2$
• D. Sum of the diagonal entries of $P^{-1}EP+F$ is equal to the sum of diaginal entries of $E+P^{-1}FP$

Answer 1

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

Sum of the diagonal entries of $P^{-1}EP+F$ is equal to the sum of diaginal entries of $E+P^{-1}FP$

Clearly, $P^2=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=1$ and
$PE=\left[\begin{array}{ccc}1& 2& 3\\ 8& 13& 18\\ 2& 3& 4\end{array}\right]$
$\Rightarrow PEP=\left[\begin{array}{ccc}1& 3& 2\\ 8& 18& 13\\ 2& 4& 3\end{array}\right]=F$


$\Rightarrow |P||E||P|=|F|$
$\Rightarrow |F|=0$ and
$|E|=\left|\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 8& 13& 18\end{array}\right|$
$R_3\to R_3-3R_2-2R_1$
$|E|=\left|\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 0& 0& 0\end{array}\right|$
$\Rightarrow |E|=0$
Now, $|EQ+PF{Q}^{-1}|$
$=|EQ+PPEP{Q}^{-1}|$
$=|E||Q+P{Q}^{-1}|$
$=0$ $(\because |E|=0)$
Also, $|EQ|+|PF{Q}^{-1}|$
$|E||Q|+|P||F||Q^{-1}|$
$=0$ $(\because |E|=0,|F|=0)$
$\therefore |EQ+PF{Q}^{-1}|=|EQ|+|PF{Q}^{-1}|$


$|(EF{)}^3|=|E{|}^3|F{|}^3=0$
$|EF{|}^2=|E{|}^2|F{|}^2=0$
$\therefore |(EF{)}^3|=|EF{|}^2$


$P^{-1}EP+F$
$\Rightarrow PEP+F$ $(\because P^2=I)$
$\Rightarrow F+F=2F$
$\Rightarrow Tr\left(2F\right)=2\times 22=44$
and
$E+P^{-1}FP$
$\Rightarrow E+P^{-1}PEPP$ $(\because F=PEP)$
$\Rightarrow E+E=2E$
$\Rightarrow Tr\left(2E\right)=2\times 22=44$