Question

$ A $ is a $ 3 \times 3 $ matrix where its first row is $ (1 \,0 \,0) $ , second row is $ (2 \,1 \,0) $ and third row is $ (3 \,2 \,1). P, Q $ and $ R $ are column matrices such that $ AP = (1 \,0 \,0)^T, AQ = (2 \,3 \,0)^T $ and $ AR = (0\, 0\, 1)^T $ . If $ P, Q $ and $ R $ are three columns of matrix $ U $ , then $ | U| = $

• A. $ 0 $
• B. $ 1 $
• C. $ 3 $
• D. $ 9 $

Answer 1

Answer: (C)

We have, $A=\left[\begin{matrix}1&0&0\\ 2&1&0\\ 3&2&1\end{matrix}\right]$
Let $P=\left[\begin{matrix}x_{1}\\ y_{1}\\ z_{1}\end{matrix}\right]$, $Q=\left[\begin{matrix}x^{2}\\ y^{2}\\ z^{2}\end{matrix}\right] $
and $R=\left[\begin{matrix}x_{3}\\ y_{3}\\ z_{3}\end{matrix}\right]$
Now, $AP=\left[\begin{matrix}x_{1}\\ 2x_{1}+y_{1}\\ 3x_{1}+2y_{1} +z_{1}\end{matrix}\right]=\left[\begin{matrix}1\\ 0\\ 0\end{matrix}\right]$
$\Rightarrow x_{1}=1, y_{1}=-2$ and $z_{1}=1$
Again, $AQ=\left[\begin{matrix}x_{2}\\ 2x_{2}+y_{2}\\ 3x_{2} 2y_{2} +z_{2}\end{matrix}\right]=\left[\begin{matrix}2\\ 3\\ 0\end{matrix}\right]$
$\Rightarrow x_{2}=2$,
$y_{2}=-1$
and $z_{2}=-4$
$\therefore AR=\left[\begin{matrix}x_{3}\\ 2x_{3} +y_{3}\\ 3x_{3}+2v_{3} +z_{3}\end{matrix}\right]=\left[\begin{matrix}0\\ 0\\ 1\end{matrix}\right]$
$\Rightarrow x_{3}=0, y_{3}=0$ and $z_{3}=1 $
So, $U=\left[\begin{matrix}1&2&0\\ -2&-1&0\\ 1&-4&1\end{matrix}\right]$
$\left|U\right|=1\left(-1+4\right)$
$=3$