Question

Find the general solutions of the systems whose augmented matrices is given?

[ 3 -4 2 0 ]
[ -9 12 -6 0 ]
[ -6 8 -4 0 ]

Answer 1

The matrix

$\left[ \begin{array}{cccc} 3 & -4 & 2 & 0\\ -9 & 12 & -6 & 0\\ -6 & 8 & -4 & 0 \end{array} \right]$

is equivalent to the matrix

$\left[ \begin{array}{cccc} 3 & -4 & 2 & 0\\ 3 & -4 & 2 & 0\\ 3 & -4 & 2 & 0 \end{array} \right]$ after dividing each element of the second row by -3 and each element of the

third row by -2. The last matrix is equivalent to the matrix

$\left[ \begin{array}{cccc} 3 & -4 & 2 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{array} \right]$ after subtraction from second and third rows the first row.

Therefore, the system is equivalent to the equation $3x_1-4x_2+2x_3=0$ which has the following infinitely many solutions:

$\begin{cases} x_1=\frac{4}{3}x_2-\frac{2}{3}x_3\\ x_2\in\mathbb R\\ x_3\in \mathbb R \end{cases}$