Chapter 7.4, Problem 6E

To determine

Whether the augmented matrixof the given system of linear equation is row equivalent to its row-reduced echelon form.

Answer to Problem 6E

The augmented matrix of the given system of linear equation is row equivalent to its row-reduced echelon form.

Explanation of Solution

Given information:

A system of linear equation is given as

  $\{\begin{array}{l}-2x+3y=5\\ 6x+7y=4\end{array}$

Concept used:

Co-efficient matrix of a system of linear equation of the form $\{\begin{array}{l}{a}_{1}x+b_{1}y=c_1\\ a_{2}x+b_{2}y=c_2\end{array}$ is given by

  $\left[\begin{array}{cc}{a}_1& b_1\\ a_2& b_2\end{array}\right]$

The augmented matrix of the system of linear equation is given by

  $\left[\begin{array}{cccc}{a}_1& b_1& ⋮& c_1\\ a_2& b_2& ⋮& c_2\end{array}\right]$

A matrix is in row echelon form if it has the following properties:

1. Any rows consisting entirely of zeros at the bottom of the matrix.
2. For each row that does not consist entirely of zeros, the first nonzero entry is 1.
3. For two successive rows, the leading 1 in the higher row is further to the left than the leading 1 in the lower row.

And a matrix in row-echelon form is in reduced row echelon form if every column that has a leading 1 has zeros in every position above and below its leading 1.

Consider the given system of equation.

  $\{\begin{array}{l}-2x+3y=5\\ 6x+7y=4\end{array}$

Now, the augmented matrix is given by

  $\left[\begin{array}{cccc}-2& 3& ⋮& 5\\ 6& 7& ⋮& 4\end{array}\right]$

Now, perform row operations as shown:

Apply $R_2\to R_2+3R_1$ .

  $\left[\begin{array}{cccc}-2& 3& ⋮& 5\\ 0& 16& ⋮& 19\end{array}\right]$

Apply $R_1\to -\frac{1}{2}{R}_1,R_2\to \frac{1}{16}{R}_2$ .

  $\left[\begin{array}{cccc}1& -\frac{3}{2}& ⋮& -\frac{5}{2}\\ 0& 1& ⋮& \frac{19}{16}\end{array}\right]$

Apply $R_1\to R_1+\frac{3}{2}{R}_2$ .

  $\left[\begin{array}{cccc}1& 0& ⋮& -\frac{23}{32}\\ 0& 1& ⋮& \frac{19}{16}\end{array}\right]$

The above is the reduced row-echelon form of the given matrix.

Therefore, the augmented matrix is row equivalent to its row-reduced echelon form.