Prentice Hall's Online Lesson Quiz Enter Web Code: aga–0408
An augmented matrix is formed by using only the coefficients from a system of equations. A vertical line is inserted to separate the two sides of the equation at the equals signs.
Since the numbers in an augmented matrix come from a system, we can do anything to this matrix that we could do the the original system. In particular, we can do the following three things to the matrix. These three things as called row transformations.
Interchange two rows.
Multiply all elements of one row by a given non-zero number.
Multiply each element of a row of the matrix by some number and add the results to the corresponding elements of another row.
The main goal of this process is to create an identity matrix on the left side of the augmented matrix where the diagonal from left to right are all ones and all other entries are zero.
It is usually a good idea to get the element 1 in row 1, column 1. In our example above that is already the case.
Next, we should get 0 in row 2, column 1. We can do this by multiplying each element of row 1 by -2, and adding the results to the elements of row 2. Then replace row 2 with your results as shown below.
Next get 0 in row 3, column 1 by subtracting row 3 from row 1 and replacing row 3 with your results.
Now focus on making row 1, column 2 zero. This is done by multiplying row 3 by -3 and adding the results to row 1.
Next we could interchange rows 2 and 3, which will give us a 1 in row 2 column 2. Remember the goal is to create a diagonal of 1 only from left to right.
Now we should make row 3, column 2 a zero by multiplying row 2 by 7 and adding the result to row 3.
Now our effort will be to make row 3, column 3 a 1. We do this by dividing row 3 by -43.
Next, we need to make row 1, column 3 a zero. This is done by multiplying row 3 by -18 and adding the results to row 1.
The final step will be to make row 2, column 3 a zero by multiplying row 3 by 8 and adding the results to row 2.
We read the solution from the columns formed x = 1, y = 0, and z = –1, written as an ordered triple (1, 0, –1).
We leave the check in the original system for the student.