Q. 24 - Precalculus Enhanced with Graphing Utilities

Question

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

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

Step-by-Step Solution

Answer

The resultant matrix is 

$\left[\begin{array}{cccc}1& 2& 1|& 8\\ 3& -5& 2|& 6\\ 0& -11& 6|& 12\end{array}\right]$

Step 1. Given Information

The given matrix and operation is 

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

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,

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

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

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

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

$$\begin{gathered} \left[\begin{array}{cccc}1& 2& 1|& 8\\ 3& -5& 2|& 6\\ -3& -6& 4|& 6\end{array}\right]\to \left[\begin{array}{cccc}1& 2& 1|& 8\\ 3& -5& 2|& 6\\ 3+(-3)& -5+(-6)& 2+4|& 6+6\end{array}\right] \\ \to \left[\begin{array}{cccc}1& 2& 1|& 8\\ 3& -5& 2|& 6\\ 0& -11& 6|& 12\end{array}\right] \end{gathered}$$