Question

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

• A. -3
• B. -1
• C. 4
• D. 2

Answer 1

Answer: (A)

Given
$\begin{bmatrix}p&q&r\end{bmatrix} \begin{bmatrix}3&4&1\\ 3&2&3\\ 2&0&2\end{bmatrix} = \begin{bmatrix}3&0&1\end{bmatrix} $
$ \Rightarrow \begin{bmatrix}3p + 3q + 2r&4p + 2q&p + 3q + 2r\end{bmatrix} = \begin{bmatrix}3&0&1\end{bmatrix}$
$ \Rightarrow \ 3p + 3q + 2r = 3 $ ...(i)
$4p + 2q = 0 \ \Rightarrow \ q = - 2p$ ...(ii)
$p + 3q + 2r = 1$ ...(iii)
On solving (i), (ii) and (iii), we get
p = 1, q = - 2, r = 3
$\therefore $ 2p + q - r = 2(1) + (- 2) - (3) = - 3.