Question

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

• A. $2$
• B. $-2$
• C. $1$
• D. $0$

Answer 1

The correct option is A

$2$

Explanation of the correct option.

Step $1$: Find the matrix $Q$.

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

Transpose the matrix.

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

Since, $Q=P{P}^T$

$$\begin{gathered} \Rightarrow Q=\left[\begin{array}{ccc}1& 2& 1\\ 1& 3& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 2& 3\\ 1& 1\end{array}\right] \\ \Rightarrow Q=\left[\begin{array}{cc}1+4+1& 1+6+1\\ 1+6+1& 1+9+1\end{array}\right] \\ \Rightarrow Q=\left[\begin{array}{cc}6& 8\\ 8& 11\end{array}\right] \end{gathered}$$

Step $2$: Find the value of $Q$.

$$\begin{gathered} Q=\left[\begin{array}{cc}6& 8\\ 8& 11\end{array}\right] \\ \Rightarrow \left|Q\right|=66-64 \\ \Rightarrow \left|Q\right|=2 \end{gathered}$$

Hence option $\left(A\right)$ is the correct option.