Question

If $p,q,r$ are $3$ real numbers satisfying the matrix equation, $\left[\begin{array}{ccc}p& q& r\end{array}\right]\left[\begin{array}{ccc}3& 4& 1\\ 3& 2& 3\\ 2& 0& 2\end{array}\right]=\left[\begin{array}{ccc}3& 0& 1\end{array}\right]$, then $2p+q-r$ equals :-

• A. $-1$
• B. $4$
• C. $-3$
• D. $2$

Answer 1

The correct option is C $-3$

$\begin{array}{c}\left[p,q,r\right]\left(\begin{array}{ccc}3& 4& 1\\ 3& 2& 2\\ 2& 0& 3\end{array}\right)=\left[3,0,1\right]\\ \left[3p+3q+2r4p+2qp+3q+2r\right]=\left[3,0,1\right]\\ 3p+3q+2r=3\to \left(i\right)\\ 4p+2q=0\to \left(ii\right)\\ p+3q+2r=1\to \left(iii\right)\\ fromeq.\left(iii\right)\\ 3q+2r=1-p\end{array}$

put the value of $3q+2rineq.\left(i\right)$

$\begin{array}{c}weget\\ 3p+1-p=3\\ 2p=2\\ p=1\end{array}$

Put the value of P in equation (ii)

$\begin{array}{c}4\left(1\right)+2q=0\\ 2q=-4\\ q=-2\end{array}$

Put the value of P and Q in equation

$\begin{array}{c}3\left(1\right)+3\left(-2\right)+2r=3\\ 3-6+2r=3\\ -6=-2r\\ r=3\\ find=2p+q-r\\ =2\left(1\right)-2-3\\ =-3\end{array}$