Chapter 7, Problem 203RE

To determine

To describe: the relation between the three elementary row operations performed on an augmented matrix corresponding to a system of equation and new system of equation after performing operation.

Explanation of Solution

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 is

1. If any row containing all elements as zero is in the bottom of the matrix.
2. Each row that does not consist entirely of zeros has the first nonzero entry as 1.
3. For successive nonzero rows the leading 1 in the higher row is further than the leading in the lower row.

And a matrix 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.

To solve a system of linear equation, the corresponding augmented matrix need to reduce in row echelon form using Gaussian elimination method.

Hence, the three elementary row operations performed on an augmented matrix corresponding to a system of equation and new system of equation after performing operation are same.