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Using Augmented Matrices to Solve Systems of Linear Equations
Elementary Row Operations
To solve the linear system algebraically, these steps could be used.
x 5y z 11
3z 12
2x 4y 2z 8
All of the following operations yield a system which is equivalent to the original. (Equivalent
systems have the same solution.)
Interchange equations 2 and 3
x 5y z 11
2x 4y 2z 8
3z 12
Multiply equation 3 by
x 5y z 11
2x 4y 2z 8
z 1
Multiply equation 2 by
x 5y z 11
x 2y z 4
z 1
Add equation 1 to 2 and replace
x 5y z 11
3y 15
z 4
equation 2 with the result
Multiply equation 2 by
x 5y z 11
y 5
z 4
Multiply equation 2 by and add it 5
x z 14
y 5
z 4
to equation 1; replace equation 1 with
the result
Add equation 3 to equation 1; replace
x 18
y 5
z 4
equation 1 with the result
The solution is (18, −5, 4).
- Operations that Produce Equivalent Systems
a) Two equations are interchanged.
b) An equation is multiplied by a nonzero constant.
c) A constant multiple of one equation is added to another equation.
Matrices
A matrix is a rectangular array of numbers written within brackets. The size of a matrix is always given
in terms of its number of rows and number of columns (in that order!). A 2 x 4 matrix has 2 rows
and 4 columns. Square matrices have the same number of rows and columns. A matrix with a single
column is called a column matrix, and a matrix with a single row is called a row matrix. A square
matrix with all elements on the main diagonal equal to 1 and all other elements equal to 0 is called an
identity matrix. The 3x3 identity matrix is.
The position of an element within a matrix is given by the row and column (in that order!) containing
the element. The element is in row 3 and column 4. 34
a
- Elementary Row Operations that Produce Row-Equivalent Matrices
a) Two rows are interchanged i j
R ↔R
b) A row is multiplied by a nonzero constant i i
kR →R
c) A constant multiple of one row is added to another row j i i
kR + R →R
(NOTE : →means "replaces")
Forming an Augmented Matrix
An augmented matrix is associated with each linear system like
x 5y z 11
3z 12
2x 4y 2z 8
The matrix to the left of the bar is called the coefficient matrix.
- Solving an Augmented Matrix
To solve a system using an augmented matrix, we must use elementary row operations to change
the coefficient matrix to an identity matrix.
Form the augmented matrix
Interchange rows 2 and 3
2 3
R ↔R
