Section 2.2 Lecture Notes
MATH 141-501
Last time, we discussed systems of linear equations.
Given a system of linear equations, there are three
things we can do that won’t change the solution of the
system.
1. Interchange (swap) any two equations.
2. Replace an equation by a nonzero constant multiple
of itself.
3. Replace any equation by the sum of that equation
and a constant multiple of any other equation.
1
Examples
1. Interchange (swap) any two equations.
2x + 3y = 4
x−y =3
2. Replace an equation by a nonzero constant multiple
of itself.
2x + 3y = 4
x−y =3
3. Replace an equation by the sum of that equation and
a constant multiple of any other equation.
2x + 3y = 4
x−y =3
2
Goal: To eliminate variables from equations so
we can see solution quickly.
Example: Solve the system of equations
2x + 4y + 6z = 22
3x + 8y + 5z = 27
−11x + 11y + 22z = 22
3
Example – continued
4
Augmented Matrices
This goes a bit more quickly if we use matrices.
Given any linear system of equations, we can convert
into an augmented matrix.
The
correspond to the
system.
of the matrix
of the linear
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Row-Reduced Form of a Matrix
We want to get our matrices into row-reduced form,
which means satisfying the following rules.
1. If a row has only zeros, it must lie below
any row having a nonzero entry.
2. The first nonzero entry of a nonzero row
must be 1 (called a leading 1).
3. In any two nonzero rows, the leading 1 in
the lower row lies to the right of the leading
1 in the upper row.
4. If a column in the coefficient matrix contains a leading 1, then the other entries in
the column are zeros.
6
Examples
1. If a row has only zeros, it must lie below any row
having a nonzero entry.
Good
Bad
2. The first nonzero entry of a nonzero row must be 1
(called a leading 1).
Good
Bad
7
Examples
3. In any two nonzero rows, the leading 1 in the lower
row lies to the right of the leading 1 in the upper row.
Good
Bad
4. If a column in the coefficient matrix contains a leading 1, then the other entries in the column are zeros.
Good
Bad
8
Practice (On Your Own)
Which of the following matrices are in row reduced
form?
If a matrix is not in row reduced form, say why.
Matrix
In Row Reduced Form?

1
0
0

1
0
0



1
0
0

0
1
0

0 0 2
1 0 3
0 1 4

0 0 4
1 0 −4 
0 0 0

1 2 3
0 0 0
4 5 6

0 0 10
3 0 1 
0 0 0

1 2 −2
0 0 3 
0 1 2
9
Solving systems using calculator
MAny calculators (including the TI-83 and TI-84) have
a built-in rref command to find row-reduced form of a
matrix.
Steps for Solving a Linear System Using Calculator (in TI-84.)
1. (Before using calculator) Move variables to one
side of each equation and constants to the other.
2. (Before using calculator)Rewrite the system so
that all variables are in the same order in each equation.
3. Create an augmented matrix representing your system of equations. (MATRIX ([2nd][x−1 ], then
EDIT menu.)
4. QUIT ([2nd][MODE])
5. Go to MATRIX again. Go to MATH menu.
6. Selection option B:rref(. Hit [ENTER.]
7. Using ALPHA, type the letter of the matrix you
created. Close parentheses, and hit ENTER.
10
The Gauss-Jordan Method
Remember that
• Interchange (swap) any two rows.
• Replace any row by a nonzero constant multiple of
itself.
• Replace any row by the sum of that row and a constant multiple of any other row.
11
Pivoting
A unit column in a matrix is a column with one
entry equal to
, and all other entries equal to
.
The process of turning a column into a unit column is
called pivoting. The entry which becomes a
called the
.




2 4 6 8
1 2 3
4
 1 1 1 1  ⇒  0 −1 −2 −3 
3 4 6 7
0 −2 −3 −5
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is
Pivoting - Example
Pivot the matrix around the circled element.
(When pivoting, show your work by writing down the
row operations that you used.)


2 1 0
 3 4 −2 
−11 6 5
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