Q. 20 - 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. 

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

Step-by-Step Solution

Answer

The resultant matrix is 

$\left[\begin{array}{cccc}1& -3& 3|& -5\\ 0& -17& 9|& -25\\ 0& -11& 13|& -9\end{array}\right]$

Step 1. Given Information

The given matrix and operation is

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

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,

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

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

$$\begin{gathered} \left[\begin{array}{cccc}1& -3& 3|& -5\\ -4& -5& -3|& -5\\ -3& -2& 4|& 6\end{array}\right]\to \left[\begin{array}{cccc}1& -3& 3|& -5\\ 4\left(1\right)+(-4)& 4(-3)+(-5)& 4\left(3\right)+(-3)|& 4(-5)+(-5)\\ -3& -2& 4|& 6\end{array}\right] \\ \to \left[\begin{array}{cccc}1& -3& 3|& -5\\ 0& -17& 9|& -25\\ -3& -2& 4|& 6\end{array}\right] \end{gathered}$$

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

$$\begin{gathered} \left[\begin{array}{cccc}1& -3& 3|& -5\\ 0& -17& 9|& -25\\ -3& -2& 4|& 6\end{array}\right]\to \left[\begin{array}{cccc}1& -3& 3|& -5\\ 0& -17& 9|& -25\\ 3\left(1\right)+(-3)& 3(-3)+(-2)& 3\left(3\right)+4|& 3(-5)+6\end{array}\right] \\ \to \left[\begin{array}{cccc}1& -3& 3|& -5\\ 0& -17& 9|& -25\\ 0& -11& 13|& -9\end{array}\right] \end{gathered}$$