Write the system of equations corresponding to each augmented matrix. Then perform the indicated row operation(s) on the given augmented matrix.

$\left[\begin{array}{cccc}1& -3& 2|& -6\\ 2& -5& 3|& -4\\ -3& -6& 4|& 6\end{array}\right],R_2=-2r_1+r_2,R_3=3r_1+r_3$

The given matrix and operation is

$\left[\begin{array}{cccc}1& -3& 2|& -6\\ 2& -5& 3|& -4\\ -3& -6& 4|& 6\end{array}\right],R_2=-2r_1+r_2,R_3=3r_1+r_3$

The given matrix has three rows. So, it represents a system of three linear equations. The 3 columns to the left of vertical bar indicate that the system has three variables.

Let the variables be x, y and z. Then system of linear equation corresponding to the given matrix is as follows,

$$\begin{gathered} x-3y+2z=-6 \\ 2x-5y+3z=-4 \\ -3x-6y+4z=6 \end{gathered}$$

Apply the first-row operation on the given augmented matrix and simplify,

$$\begin{gathered} \left[\begin{array}{cccc}1& -3& 2|& -6\\ 2& -5& 3|& -4\\ -3& -6& 4|& 6\end{array}\right]\to \left[\begin{array}{cccc}1& -3& 2|& -6\\ -2\left(1\right)+2& -2(-3)+(-5)& -2\left(2\right)+3|& -2(-6)+(-4)\\ -3& -6& 4|& 6\end{array}\right] \\ \to \left[\begin{array}{cccc}1& -3& 2|& -6\\ 0& 1& -1|& 8\\ -3& -6& 4|& 6\end{array}\right] \end{gathered}$$

Now, apply the second-row operation on the obtained matrix and simplify,

localid="1647504335664" $$\begin{gathered} \left[\begin{array}{cccc}1& -3& 2|& -6\\ 0& 1& -1|& 8\\ -3& -6& 4|& 6\end{array}\right]\to \left[\begin{array}{cccc}1& -3& 2|& -6\\ 0& 1& -1|& 8\\ 3\left(1\right)+(-3)& 3(-3)+(-6)& 3\left(2\right)+4|& 3(-6)+6\end{array}\right] \\ \to \left[\begin{array}{cccc}1& -3& 2|& -6\\ 0& 1& -1|& 8\\ 0& -15& 10|& -12\end{array}\right] \end{gathered}$$