Math-2 Honors Matrix Gaussian Elimination Lesson 11.3

Where does a “matrix” come from? Expressions Not equations since no equal sign 2x – 3y 7x + 2y 2x – 3y = 8 7x + 2y = 2 Equations Are equations since there is an equal sign

Where does a “matrix” come from? Matrix of coefficients 2x – 3y 7x + 2y 2 –3 7 2 2 –3 8 7 2 2 2x – 3y = 8 7x + 2y = 2

Big Picture 3 5 3 1 0 -4 -1 2 10 0 1 3 x + 0y = -4 3x + 5y = 3 We will perform “row operations” to turn the left side matrix into the matrix on the right side. 3 5 3 -1 2 10 1 0 -4 0 1 3 x + 0y = -4 0x + 1y = 3 3x + 5y = 3 -x + 2y = 10 x = -4 y = 3

1 0 -4 0 1 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. 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. 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 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. # in 2nd row # in 1st row

1st step: we still want a zero in the bottom left corner. -1 2 10 3 5 3 5 3 + 3(-1 2 10) 5 3 +3(-1) +3(10) +3(2) 11 33 New Row 2

1st step: we still want a zero in the bottom left corner. -1 2 10 3 5 3 5 3 + 3(-1 2 10) -1 2 10 0 11 33 5 3 +3(-1) +3(10) +3(2) 11 33 New Row 2

2nd step: we want a one in 2nd position of the 2nd row. -1 2 10 0 11 33 11 33 11 11 11 1 3 New Row 2

2nd step: we want a one in 2nd position of the 2nd row. -1 2 10 0 11 33 11 33 11 11 11 1 3 -1 2 10 0 1 3 New Row 2

-1 2 10 0 1 3 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).

-1 2 10 0 1 3 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). 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. # in 1st row # in 2nd row

3rd step: we want a zero in 2nd position of the 1st row. -1 2 10 0 1 3 -1 2 10 -3(0 1 3) -1 2 10 -2(0) -2(1) -2(3) -1 4 New Row 1

3rd step: we want a zero in 2nd position of the 1st row. -1 2 10 0 1 3 -1 2 10 -3(0 1 3) -1 2 10 -1 0 4 0 1 3 -2(0) -2(1) -2(3) -1 4 New Row 1

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 4 0 1 3 1 -4 New Row 2 1 0 -4 0 1 3

Look at the circular pattern Don’t freak out: this goes faster than you think. Look at the circular pattern -1 2 10 3 5 3

Look at the circular pattern 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

Look at the circular pattern 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

Look at the circular pattern 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 -1 2 10 0 1 3

Look at the circular pattern 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 -1 2 10 0 1 3 -1 2 10 0 1 3

Look at the circular pattern Don’t freak out: this goes faster than you think. Look at the circular pattern -1 2 10 3 5 3 -1 0 4 0 1 3 -1 2 10 0 11 33 -1 2 10 0 11 33 -1 2 10 0 1 3 -1 2 10 0 1 3

Look at the circular pattern Don’t freak out: this goes faster than you think. Look at the circular pattern -1 2 10 3 5 3 -1 0 4 0 1 3 -1 0 4 0 1 3 -1 2 10 0 11 33 -1 2 10 0 11 33 -1 2 10 0 1 3 -1 2 10 0 1 3

Look at the circular pattern Don’t freak out: this goes faster than you think. Look at the circular pattern -1 2 10 3 5 3 1 0 -4 0 1 3 -1 0 4 0 1 3 -1 0 4 0 1 3 -1 2 10 0 11 33 -1 2 10 0 11 33 -1 2 10 0 1 3 -1 2 10 0 1 3