Math-2 Honors
Matrix
Gaussian Elimination
Lesson 11.3
Where does a “matrix” come
from?
2x – 3y
7x + 2y
2x – 3y = 8
7x + 2y = 2
Expressions
Not equations since
no equal sign
Equations
Are equations since
there is an equal sign
Where does a “matrix” come
from?
Matrix of
coefficients
2x – 3y
2
–3
7x + 2y
7
2
2x – 3y = 8
2
–3
8
7x + 2y = 2
7
2
2
Big Picture
We will perform “row operations” to turn the left
side matrix into the matrix on the right side.
3
5
3
-1
2
10
3x + 5y = 3
-x + 2y = 10

1
0
-4
0
1
3

x + 0y = -4
0x + 1y = 3
x = -4
y= 3
We call this reduced row eschelon form.
1
0
-4
0
1
3
1’s on the main diagonal
0’s above/below the main diagonal
How do I do that?
Similar to elimination, we add multiples of one
row to another row.
BUT, unlike elimination, we only change one
row at a time and we end up with the same
number of rows that we started with.
Some important principles about systems of equations.
3x  6 y  6
x  4 y  12
x  4 y  12
3x  6 y  6
Are the graphs of these two systems different
from each other?
Principle 1: you can exchange rows of a matrix.
Some important principles about systems of equations.
3x  6 y  6
x  2y  2
x  4 y  12
x  4 y  12
Are the graphs of these two systems different
from each other?
Principle 2: you multiply (or divide) any row by a
number and it won’t change the graph (or the matrix)
1st step: we want a zero in the bottom left corner.
3
5
3
-1 2 10
But, you will see later that this will
be easier if the top left number is a
one or a negative one.
Swap rows.
-1 2 10
3 5
3
1st step: we still want a zero in the bottom left corner.
-1 2 10
3 5
3
Forget about all the numbers but
the 1st column (for a minute).
1st step: we still want a zero in the bottom left corner.
-1 2 10
3 5
3
3  3(1)  0
# in
2nd row
# in
1st row
Forget about all the numbers but
the 1st column (for a minute).
What multiple of the 1st row should
we add or subtract from row 2 to
turn the 3 into a zero?
This gives us the pattern of what to
do to each other number in row 2.
R2  3 * R1
1st step: we still want a zero in the bottom left corner.
-1 2 10
3 5
3
R2  3 * R1
3
+3(-1)
3
5 3
+ 3(-1 2 10)
5
+3(2)
0
New Row 2
11
3
+3(10)
33
1st step: we still want a zero in the bottom left corner.
-1 2 10
3 5
3
-1 2 10
0 11 33
R2  3 * R1
3
+3(-1)
3
5 3
+ 3(-1 2 10)
5
+3(2)
0
New Row 2
11
3
+3(10)
33
2nd step: we want a one in 2nd position of the 2nd row.
-1 2 10
0 11 33
0
R2 11 11
11
11
11
0
1
3
New Row 2
33
2nd step: we want a one in 2nd position of the 2nd row.
-1 2 10
0 11 33
-1 2 10
0 1
3
0
R2 11 11
11
11
11
0
1
3
New Row 2
33
3rd step: we want a zero in 2nd position of the 1st row.
-1 2 10
0 1
3
Forget about all the numbers but
the 2nd column (for a minute).
3rd step: we want a zero in 2nd position of the 1st row.
-1 2 10
0 1
3
2  2(1)  0
# in
1st row
# in
2nd row
Forget about all the numbers but
the 2nd column (for a minute).
What multiple of the 2nd row should
we add or subtract from the 1st row
to turn the 2 into a zero?
This gives us the pattern of what to
do to each other number in row 1.
R1  2 * R2
3rd step: we want a zero in 2nd position of the 1st row.
-1 2 10
0 1
-1
3 R1  2 * R2
-3(0
-1
2
-2(0) -2(1)
-1
New Row 1
0
2 10
1
3)
10
-2(3)
4
3rd step: we want a zero in 2nd position of the 1st row.
-1 2 10
0 1
-1
3 R1  2 * R2
-1 0 4
0 1 3
-3(0
-1
2
-2(0) -2(1)
-1
New Row 1
0
2 10
1
3)
10
-2(3)
4
4th step: we want a one in the top left corner.
-1 0 4
0 1 3
4th step: we want a one in the top left corner.
-1 0
0 1
4
3
1 0
-4
0 1
3
(1) * R1
1
0
New Row 2
-4
Don’t freak out: this goes faster than you think.
Look at the circular pattern
-1 2 10
3 5
3
Don’t freak out: this goes faster than you think.
Look at the circular pattern
-1 2 10
3 5
3
-1 2 10
0 11 33
Don’t freak out: this goes faster than you think.
Look at the circular pattern
-1 2 10
3 5
3
-1 2 10
0 11 33
-1 2 10
0 11 33
Don’t freak out: this goes faster than you think.
Look at the circular pattern
-1 2 10
3 5
3
-1 2 10
0 11 33
-1 2 10
-1 2 10
0 11 33
0 1
3
Don’t freak out: this goes faster than you think.
Look at the circular pattern
-1 2 10
3 5
3
-1 2 10
0 11 33
-1 2 10
-1 2 10
-1 2 10
0 11 33
0 1
0 1
3
3
Don’t freak out: this goes faster than you think.
Look at the circular pattern
-1 2 10
3 5
3
-1 2 10
-1 0 4
0 11 33
0 1 3
-1 2 10
-1 2 10
-1 2 10
0 11 33
0 1
0 1
3
3
Don’t freak out: this goes faster than you think.
Look at the circular pattern
-1 2 10
-1 0 4
3 5
3
0 1 3
-1 2 10
-1 0 4
0 11 33
0 1 3
-1 2 10
-1 2 10
-1 2 10
0 11 33
0 1
0 1
3
3
Don’t freak out: this goes faster than you think.
Look at the circular pattern
-1 2 10
1 0
-4
-1 0 4
3 5
0 1
3
0 1 3
3
-1 2 10
-1 0 4
0 11 33
0 1 3
-1 2 10
-1 2 10
-1 2 10
0 11 33
0 1
0 1
3
3