Question

Consider a linear system of three equations with three unknowns. We are told that the system has a unique solution. What does the reduced row-echelon form of the coefficient matrix of this system look like? Explain your answer.

Short answer

The reduced row-echelon form of a linear system of 3 equations with 3 unknowns is,

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Step-by-step solution

Consider the linear system of equations

A matrix in reduced row-echelon form is used to solve systems of linear equations. There are four prerequisites for the reduced row echelon form:

• The number 1 is the first non-zero integer in the first row (the leading entry).
• The second row begins with the number 1, which is more to the right than the first row's leading item. The number 1 must be further to the right in each consecutive row.
• Each row's first item must be the sole non-zero number in its column.
• Any rows that are not zero are pushed to the bottom of the matrix.

Consider a linear system of 3 equations as,

$\left[\begin{array}{c}{a}_{1}x+b_{1}y+c_{1}z=0\\ a_{2}x+b_{2}y+c_{2}z=0\\ a_{3}x+b_{3}y+c_{3}z=0\end{array}\right]$

The matrix form of the linear system of equations is,

Consider the reduced row-echelon form

The reduced row-echelon form of the matrix is,

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

There are no free variables present in the matrix, thus, the system will have unique solution.