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 $2p-q+r$ equaIs :

Answer 1

$\left[pqr\right]\left[\begin{array}{ccc}3& 4& 1\\ 3& 2& 3\\ 2& 0& 2\end{array}\right]=\left[301\right]$

$3p+3q+2r=3$.....(i)

$4p+2q+0r=0$.....(ii)

$1p+3q+2r=1$.....(iii)

Substract (iii) from (i)

$3p-1p+3q-3q+2r-2r=3-1$

$2p=2\Rightarrow p=1$

Substituting in equation (ii)

$4\left(1\right)+2q=0$

$2q=-4\Rightarrow q=-2$

$1\left(1\right)+3(-2)+2r=1$

$2r=6\Rightarrow r=3$

$2p-q+r$

$2\left(1\right)-(-2)+3=7$.