If
$x_{1} = 3 y_{1} + 2 y_{2} - y_{3}, y_{1} = z_{1} - z_{2} + z_{3} x_{2} = - y_{1} + 4 y_{2} + 5 y_{3}, y_{2} = z_{2} + 3 z_{3} x_{3} = y_{1} - y_{2} + 3 y_{3}, y_{3} = 2 z_{1} + z_{2}$
express $x_{1},$ in terms of $z_{1}, z_{2}, z_{3}$ .
we get $x_{1} = z_{1} - 2 z_{2} + k z_{3}$ , what is the value of k?

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Solution

Since,
$x_{1} = 3 y_{1} + 2 y_{2} - y_{3} [x_{1}] = [\begin{matrix} 3 & 2 & - 1 \end{matrix}] ⎡ ⎢ ⎣ \begin{matrix} y_{1} y_{2} y_{3} \end{matrix} ⎤ ⎥ ⎦$
Putting the values of $y_{1}, y_{2}, y_{3}$ we get
$= [\begin{matrix} 3 & 2 & - 1 \end{matrix}] ⎡ ⎢ ⎣ \begin{matrix} z_{1} - z_{2} + z_{3} 0 + z_{2} + 3 z_{3} 2 z_{1} + z_{2} + 0 \end{matrix} ⎤ ⎥ ⎦$
$= [\begin{matrix} 3 & 2 & - 1 \end{matrix}] ⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 & 1 0 & 1 & 3 2 & 1 & 0 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} z_{1} z_{2} z_{3} \end{matrix} ⎤ ⎥ ⎦$
$= [\begin{matrix} 3 + 0 - 2 & - 3 + 2 - 1 & 3 + 6 + 0 \end{matrix}] ⎡ ⎢ ⎣ \begin{matrix} z_{1} z_{2} z_{3} \end{matrix} ⎤ ⎥ ⎦$
$= [\begin{matrix} 1 & - 2 & 9 \end{matrix}] ⎡ ⎢ ⎣ \begin{matrix} z_{1} z_{2} z_{3} \end{matrix} ⎤ ⎥ ⎦$
$[x_{1}] = [z_{1} - 2 z_{2} + 9 z_{3}]$
$∴ x_{1} = z_{1} - 2 z_{2} + 9 z_{3}$
$∴ k = 9$

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If x1=3y1+2y2−y3,y1=z1−z2+z3x2=−y1+4y2+5y3,y2=z2+3z3x3=y1−y2+3y3,y3=2z1+z2express x1, in terms of z1, z2, z3.we get x1=z1−2z2+kz3, what is the value of k?