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

$\left[\begin{array}{cccc}5& -3& 1|& -2\\ 2& -5& 6|& -2\\ -4& 1& 4|& 6\end{array}\right],R_1=-2r_2+r_1,R_3=2r_2+r_3$

The given matrix and operation is

$\left[\begin{array}{cccc}5& -3& 1|& -2\\ 2& -5& 6|& -2\\ -4& 1& 4|& 6\end{array}\right],R_1=-2r_2+r_1,R_3=2r_2+r_3$

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,

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

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

$$\begin{gathered} \left[\begin{array}{cccc}5& -3& 1|& -2\\ 2& -5& 6|& -2\\ -4& 1& 4|& 6\end{array}\right]\to \left[\begin{array}{cccc}-2\left(2\right)+5& -2(-5)+(-3)& -2\left(6\right)+1|& -2(-2)+(-2)\\ 2& -5& 6|& -2\\ -4& 1& 4|& 6\end{array}\right] \\ \to \left[\begin{array}{cccc}1& 7& -11|& 2\\ 2& -5& 6|& -2\\ -4& 1& 4|& 6\end{array}\right] \end{gathered}$$

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

localid="1647504615818" $$\begin{gathered} \left[\begin{array}{cccc}1& 7& -11|& 2\\ 2& -5& 6|& -2\\ -4& 1& 4|& 6\end{array}\right]\to \left[\begin{array}{cccc}1& 7& -11|& 2\\ 2& -5& 6|& -2\\ 2\left(2\right)+(-4)& 2(-5)+1& 2\left(6\right)+4|& 2(-2)+6\end{array}\right] \\ \to \left[\begin{array}{cccc}1& 7& -11|& 2\\ 2& -5& 6|& -2\\ 0& -9& 16|& 2\end{array}\right] \end{gathered}$$