If p,q,r are 3 real numbers satisfying the matrix equation pmatrixp&q&rpmatrixpmatrix3&4&1…

Question

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

Accepted Answer

Expanding: $3p+3q+2r=3$, $4p+2q=0$, $p+3q+2r=1$.

From the second: $q=-2p$. Substituting into the first and third:
$$3p-6p+2r=3\Rightarrow-3p+2r=3,\qquad p-6q\ \text{(recheck)}: p+3(-2p)+2r=1\Rightarrow-5p+2r=1.$$

Subtracting: $2p=2\Rightarrow p=1$, so $q=-2$, and $-3(1)+2r=3\Rightarrow r=3$.

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

Solutions · 1