Linear Systems and Augmented Matrices

What is an augmented matrix?  An augmented matrix is essentially two matrices put together.  In the case of a system of linear equations, it’s composed of the coefficients of the equations and their constant terms.  Essentially, it’s just the system of equations without the variables or the plus, minus, or equals signs.

Translating a System to a Matrix 1. In each equation, put your variable terms on one side, constants on the other. 2. Find your coefficients. 3. Form a matrix using the coefficients 4. Augment the matrix by adding the constant terms to the right side, separated by a dotted line.

Why Matrices?  Matrices are a compact way of presenting information.  Matrices are easy to work with using elementary row operations.

Example Let’s go through the process of converting a system of linear equations to an augmented matrix for this system of equations: 2x = 3 – 5y -7x + y = 0

Step 1: Organize Your Equations  In general, when you have two or three variables, you want your equations to be in the form ax + by = c or ax + by + cz = d, respectively. a, b, c, and d are constants and x, y, and z are your variables.  Make sure all your equations have the variables in the same order!  In our system, all we have to do is add 5y to both sides of the first equation, leaving 2x + 5y = 3 -7x + y = 0

Step 2: Find the Coefficients  The coefficients are the constants that are multiplied by your variables.  Keep any negative signs.  If a variable is not present in a particular equation, its coefficient is zero.  In our system of equations, 2x + 5y = 3 -7x + y = 0 Our coefficients are 2 and 5 in the first equation, -7 and 1 in the second.

Step 3: Put the Coefficients in a Matrix  Each equation forms a row of the matrix.  Each variable forms a column.  For our system of linear equations, 2x + 5y = 3 -7x + y = 0 The matrix will be

Step 4: Augment the Matrix  Draw a dotted line down the right side of your matrix.  Add the constant term of each equation to the right of the line.  In our system of equations, the constants in the first and second equations are 3 and 0, respectively. Thus, the final augmented matrix will look like this: