DIalogue about the augmented matrix of a system of linear equations

This short dialogue started with someone posting on LinkedIn group Math, Math Education, Math Culture a link to the following blogpost:

Matrices are a common tool used in algebra. They provide a way to deal with equations that have commonly held variables. In this post, we learn some of the basics of developing matrices.

From Equation to Matrix

Using a matrix involves making sure that the same variables and constants are all in the same column in the matrix. This will allow you to do any elimination or substitution you may want to do in the future. Below is an example

Above we have a system of equations to the left and an augmented matrix to the right. If you look at the first column in the matrix it has the same values as the x variables in the system of equations (2 & 3). This is repeated for the y variable (-1 & 3) and the constant (-3 & 6).

The number of variables that can be included in a matrix is unlimited. Generally,  when learning algebra, you will commonly see 2 & 3 variable matrices. The example above is a 2 variable matrix below is a three-variable matrix.

If you look closely you can see there is nothing here new except the z variable with its own column in the matrix.

 

Row Operations 

When a system of equations is in an augmented matrix we can perform calculations on the rows to achieve an answer. You can switch the order of rows as in the following.

You can multiply a row by a constant of your choice. Below we multiple all values in row 2 by 2. Notice the notation in the middle as it indicates the action performed.

You can also add rows together. In the example below row 1 and row 2, are summed to create a new row 1.

You can even multiply a row by a constant and then sum it with another row to make a new row. Below we multiply row 2 by 2 and then sum it with row 1 to make a new row 1.

The purpose of row operations is to provide a way to solve a system of equations in a matrix. In addition, writing out the matrices provides a way to track the work that was done. It is easy to get confused even the actual math is simple

Conclusion

System of equations can be difficult to solve. However, the use of matrices can reduce the computational load needed to solve them. You do need to be careful with how you modify the rows and columns and this is where the use of row operations can be beneficial

What follows is a short exchange of comments:

AB: What is missed in this discussion of augmented [matrix] is the key issue: why do row operations lead to a solution of the original system of simultaneous linear equation? This crucial fact is stated, but not explained in any way.

Response: Well i wasn’t trying to be that deep. My goal is to support students who are learning algebra. Perhaps you can explain here why row operations work.

AB: It is a really simple but fundamental fact: if you have a system of simultaneous linear equations, then the following operations:

* swapping two equations
* adding a multiple of one equation to another equation
* multiplying an equation by a non-zero number

do not change (the set of) solutions; it is useful to start discussion of system of linear equations by proving this statement and emphasising that these operations are reversible; the proof crucially depends on this observation. Row operations on the augmented matrix are operations on the system of equations. If we do some row operations on the augmented matrix and then write a system of equations corresponding to the resulting matrix, the new system of equation in equivalent to the original one, that is, the two systems have exactly the same solutions.