Matrices Pt 3

Augmented Matrices
An augmented matrix is a matrix made of several linear equations that make up a system of equations. Each row of the matrix represents a equations within the system, whereas each column of the matrix represents a variable wtihin that equation.

We can see this with the above example, where we have two equations within a system and a matrix representation of that system. It also shows how the columns represents each variable or constant within the equation. For the x variables we have both 2 & 3, for y variables we have 5 & 4 and for the constant we have 10 & 24. This means we can write equations without having to write the x or y letters, or any other letters that might be used.

When converting a system of equations to a matrix, we need to make sure that each variable and constant is in line with each other but also if a variable or constant doesn't exist in one equation of the system then it defaults to 0 within the matrix at the location where the value would have gone.

Matrix Row Operators
There are three types of matrix row operators:

1. Switch any two rows
2. Multiply a row by a nonzero constant
3. Add one row to another

You will know to switch two rows of a matrix when given a question like R1 ↔ R2, which means swap row 1 with row 2. It is important to note that you can also subtract if needed.

You will know to multiply a row of a matrix by a nonzero constant when given a question like 3R1 → R1, which means multiply row 1 by 3. It is also important to note that you can essentially divide by a nonzero constant as well by using fractions like 1/2.

You will know to multiply a row of a matrix by a nonzero constant when given a question like R2 + R3 → R2, which means add row 2 to row 3 and enter the answer in row 2.

Each row operator could also be represented in this format:

On the left side we have the original matrix, in the center is the row operator and the direction being to the final matrix, which we can see on the right side.

It's also worth noting that they can be combined.

Notice how the 2R3 part only applies to the addition in row 1 and is not used to modify row 3.

As mentioned before augmented matrices represent a system of equations. By using row operators we can get equivelent systems of equations. For example by using the mutliplaction by a nonzero constant we can eliminate a variable from an equation.

This is our original system of equations and the augmented matrix. If we carried out -2R1 → R1, we get:

This then allows us to use the row addition operator; R1 + R2 → R2, which gives us:

In other words, we get the final equation -1y = -4, which we can use to solve for y. So y = 4, which can then be subsituted in any of the equations to solve for x.

Reduced Row Echelon Form
You can solve a system of equations by converting the augmented matrix into the reduced row echelon form. This form is were the diagonal components of a matrix are valued at 1 and every other component is valued at 0, except for any constants. Think of it being a combination of the identity matrix and an augmented matrix. Below is an example, where a, b & c are the constants.

To get to the reduced row echelon form, can use the row operators from earlier.

Once a augmented matrix has been converted we can solve the system of equations. So using the above example we can see that x = 5, y = -1 & z = -1 and if we subsitute them back into the original equations we can see that everything is correct.

1x + 1y + 1z = 3

1(5) + 1(-1) + 1(-1) = 3

5 - 1 -1 = 3