Question

If $P=\left[\begin{array}{ccc}1& 2& 1\\ 1& 3& 1\end{array}\right],Q=P{P}^T$, then the value of the determinant of $Q$ is

• A. $2$
• B. $-2$
• C. $1$
• D. $0$

Answer 1

The correct option is A $2$

Given,

$P=\left[\begin{array}{ccc}1& 2& 1\\ 1& 3& 1\end{array}\right]$

$\therefore Q=P{P}^T=\left[\begin{array}{ccc}1& 2& 1\\ 1& 3& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 2& 3\\ 1& 1\end{array}\right]$

$=\left[\begin{array}{cc}1\times 1+2\times 2+1\times 1& 1\times 1+2\times 3+1\times 1\\ 1\times 1+3\times 2+1\times 1& 1\times 1+3\times 3+1\times 1\end{array}\right]$

$=\left[\begin{array}{cc}1+4+1& 1+6+1\\ 1+6+1& 1+9+1\end{array}\right]=\left[\begin{array}{cc}6& 8\\ 8& 11\end{array}\right]$

$\therefore$ The determinant of $Q=\left[\begin{array}{cc}6& 8\\ 8& 11\end{array}\right]$

$=66-64=2$