Question

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

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

Short answer

The resultant matrix is

$\left[\begin{array}{cccc}1& -3& -4|& -6\\ 0& 13& 30|& 30\\ 0& -2& 0|& 0\end{array}\right]$

Step-by-step solution

Step 1. Given Information 

The given matrix and operation is

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

Step 2. Explanation   

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,

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

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

$$\begin{gathered} \left[\begin{array}{cccc}1& -3& -4|& -6\\ 6& -5& 6|& -6\\ -1& 1& 4|& 6\end{array}\right]\to \left[\begin{array}{cccc}1& -3& -4|& -6\\ -6\left(1\right)+6& -6(-3)+(-5)& -6(-4)+6|& -6(-6)+(-6)\\ -1& 1& 4|& 6\end{array}\right] \\ \to \left[\begin{array}{cccc}1& -3& -4|& -6\\ 0& 13& 30|& 30\\ -1& 1& 4|& 6\end{array}\right] \end{gathered}$$

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

localid="1647504445199" $$\begin{gathered} \left[\begin{array}{cccc}1& -3& -4|& -6\\ 0& 13& 30|& 30\\ -1& 1& 4|& 6\end{array}\right]\to \left[\begin{array}{cccc}1& -3& -4|& -6\\ 0& 13& 30|& 30\\ 1+(-1)& -3+1& -4+4|& -6+6\end{array}\right] \\ \to \left[\begin{array}{cccc}1& -3& -4|& -6\\ 0& 13& 30|& 30\\ 0& -2& 0|& 0\end{array}\right] \end{gathered}$$