After performing Gaussian elimination a few times, you probably noticed that you spent most of the time concentrating on the coefficients and simply recorded the variables as place holders. Based on this observation, we will introduce a shorthand description of linear systems.
When writing a linear system, we always write the variables in the same order in each equation. We then construct an augmented matrix by simply forgetting about the variables and recording the numerical data in a rectangular array. For instance, the system of equations below has the following augmented matrix
The vertical line reminds us where the equals signs appear in the equations. Entries in the matrix to the left of the vertical line correspond to coefficients of the equations. We sometimes choose to focus only on the coefficients of the system in which case we write the coefficient matrix as
The three operations we perform on systems of equations translate naturally into operations on matrices. For instance, the replacement operation that multiplies the first equation by 2 and adds it to the second may be performed by multiplying the first row of the augmented matrix by 2 and adding it to the second row:
The symbol \(\sim\) between the matrices indicates that the two matrices are related by a sequence of scaling, interchange, and replacement operations. Since these operations act on the rows of the matrices, we say that the matrices are row equivalent. Notice that the linear systems corresponding to two row equivalent augmented matrices have the same solution space.
Activity4.4.1.Augmented matrices and solution spaces.
Make a choice for the names of the variables and write the corresponding linear system. Does the system have exactly one solution, infinitely many solutions, or no solutions?
Answer.
There is a single solution \((3,-1,0)\text{.}\)
There is a single solution \((3,0)\text{.}\)
There are no solutions.
This system has three equations in five variables, and there are infinitely many solutions.
The third equation is satisfied for any values of \(x\) and \(y\text{.}\) Therefore, we see that the only solution to the system is \((x,y) = (3,0)\text{.}\)
Since the third equation is not satisfied for any values of \(x\) and \(y\text{,}\) there are no solutions to the system.
The system corresponding to this augmented matrix has three equations and five variables. Our first guess is there are infinitely many solutions. If we write out the equations corresponding to the augmented matrix, we find
since the third row of the augmented matrix does not restrict the solution space. From here, we see that there are infinitely many solutions: if we make any choice for the variables \(x_2\text{,}\)\(x_4\text{,}\) and \(x_5\text{,}\) we can find values for \(x_1\) and \(x_3\) that give a solution.
