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

We begin with the matrix equation

$$[p \ q \ r]\begin{bmatrix} 3 & 4 & 1 \\ 3 & 2 & 3 \\ 2 & 0 & 2 \end{bmatrix}= [3 \ 0 \ 1].$$

For any row vector $$[a \ b \ c]$$ and square matrix $$\begin{bmatrix} x_{11}&x_{12}&x_{13}\\x_{21}&x_{22}&x_{23}\\x_{31}&x_{32}&x_{33}\end{bmatrix},$$ the rule of matrix multiplication tells us that the resulting row vector is obtained by taking the dot-product of $$[a \ b \ c]$$ with each column of the matrix. Hence

$$[p \ q \ r] \begin{bmatrix} 3 & 4 & 1 \\ 3 & 2 & 3 \\ 2 & 0 & 2 \end{bmatrix} =\bigl(p\cdot3+q\cdot3+r\cdot2,\; p\cdot4+q\cdot2+r\cdot0,\; p\cdot1+q\cdot3+r\cdot2\bigr).$$

According to the given equation, this row vector equals $$[3 \ 0 \ 1].$$ So we equate the corresponding components:

$$\begin{aligned} 3p+3q+2r &= 3,\\ 4p+2q &= 0,\\ p+3q+2r &= 1. \end{aligned}$$

From the second equation we solve directly for $$q$$:

$$4p+2q=0 \;\;\Longrightarrow\;\; 2p+q=0 \;\;\Longrightarrow\;\; q=-2p.$$

We now substitute $$q=-2p$$ into the first and third equations.

Substituting in $$3p+3q+2r=3$$ gives

$$3p+3(-2p)+2r=3 \;\;\Longrightarrow\;\; 3p-6p+2r=3 \;\;\Longrightarrow\;\; -3p+2r=3. \quad -(1)$$

Substituting in $$p+3q+2r=1$$ gives

$$p+3(-2p)+2r=1 \;\;\Longrightarrow\;\; p-6p+2r=1 \;\;\Longrightarrow\;\; -5p+2r=1. \quad -(2)$$

We now have the simpler system

$$\begin{aligned} -3p+2r &= 3,\\ -5p+2r &= 1. \end{aligned}$$

Subtracting the first equation from the second eliminates $$r$$:

$$(\,-5p+2r) - (\,-3p+2r) = 1-3 \;\;\Longrightarrow\;\; -5p+2r+3p-2r = -2 \;\;\Longrightarrow\;\; -2p = -2,$$

so

$$p = 1.$$

Putting $$p=1$$ into $$q=-2p$$ gives

$$q=-2.$$

Substituting $$p=1$$ in equation (1), $$-3(1)+2r=3$$ leads to

$$-3+2r=3 \;\;\Longrightarrow\;\; 2r=6 \;\;\Longrightarrow\;\; r=3.$$

We have now found

$$p=1,\qquad q=-2,\qquad r=3.$$

The required expression is $$2p+q-r,$$ so we compute

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

Hence, the correct answer is Option A.