Question

Sol. [Key Concept: Recover the rows of M from the three given products, then solve $M\ \lbrack x,y,z\rbrack^{T} = \lbrack 1,7,11\rbrack^{T}$ for $x + y + z$.]

Let $M = \begin{bmatrix} a_{1} & a_{2} & a_{3} \\ {\text{ }b}_{1} & {\text{ }b}_{2} & {\text{ }b}_{3} \\ c_{1} & c_{2} & c_{3} \end{bmatrix}$

then $M\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$

$${\Rightarrow a_{1} = 1,{\text{ }b}_{1} = 2,c_{1} = 3 }{M\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix} \Rightarrow a_{2} = 0,{\text{ }b}_{2} = 1,c_{2} = 2}$$

$$ {\& M\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} - 1 \\ 1 \\ 1 \end{pmatrix} \Rightarrow a_{3} = - 1,{\text{ }b}_{3} = 1,c_{3} = 1}$$

Now $M\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 1 \\ 7 \\ 11 \end{pmatrix}$

$$\Rightarrow x + 0y - z = 1$$

$$\begin{matrix} & 2x + y + z = 7 \\ & 3x + 2y + z = 11 \end{matrix}$$

on solving $x = 2,y = 2,z = 1$

$$\therefore x + y + z = 5 $$