1. Math-2 Honors Matrix Gaussian Elimination
Lesson 11.3

4. 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

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

6. 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.
x + 0y = -4
0x + 1y = 3
3x + 5y = 3
-x + 2y = 10
x = -4
y = 3

7. 1 0 -4 0 1 3 We call this reduced row eschelon form.
1’s on the main diagonal
0’s above/below the main diagonal

8. 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.

9. 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.

10. 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)

11. 1st step: we want a zero in the bottom left corner.
But, you will see later that this will be easier if the top left number is a one or a negative one.
Swap rows.

12. 1st step: we still want a zero in the bottom left corner.
Forget about all the numbers but the 1st column (for a minute).

13. 1st step: we still want a zero in the bottom left corner.
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

14. 1st step: we still want a zero in the bottom left corner.
+ 3( )
+3(-1)
+3(10)
+3(2) 11 33
New Row 2

15. 1st step: we still want a zero in the bottom left corner.
+ 3( )
+3(-1)
+3(10)
+3(2) 11 33
New Row 2

16. 2nd step: we want a one in 2nd position of the 2nd row. 11 33 11 11 11 1 3
New Row 2

17. 2nd step: we want a one in 2nd position of the 2nd row. 11 33 11 11 11 1 3
New Row 2

18. -1 2 10 0 1 3 3rd step: we want a zero in 2nd position of the 1st row.
Forget about all the numbers but the 2nd column (for a minute).

19. -1 2 10 0 1 3 3rd step: we want a zero in 2nd position of the 1st 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.
# in
1st row
# in
2nd row

20. 3rd step: we want a zero in 2nd position of the 1st row.
-3( )
-2(0)
-2(1)
-2(3) -1 4
New Row 1

21. 3rd step: we want a zero in 2nd position of the 1st row.
-3( )
-2(0)
-2(1)
-2(3) -1 4
New Row 1

22. 4th step: we want a one in the top left corner.

23. 4th step: we want a one in the top left corner. 1 -4
New Row 2

24. Look at the circular pattern
Don’t freak out: this goes faster than you think.
Look at the circular pattern

25. Look at the circular pattern
Don’t freak out: this goes faster than you think.
Look at the circular pattern

26. Look at the circular pattern
Don’t freak out: this goes faster than you think.
Look at the circular pattern

27. Look at the circular pattern
Don’t freak out: this goes faster than you think.
Look at the circular pattern

28. Look at the circular pattern
Don’t freak out: this goes faster than you think.
Look at the circular pattern

29. Look at the circular pattern
Don’t freak out: this goes faster than you think.
Look at the circular pattern

30. Look at the circular pattern
Don’t freak out: this goes faster than you think.
Look at the circular pattern

31. Look at the circular pattern
Don’t freak out: this goes faster than you think.
Look at the circular pattern

32. Basic Operations: Addition
2 –3 5
(–3+1) + =

33. Basic Operations: Addition
2 –3 5 -2 + =

34. Basic Operations: Addition
2 –3 5 -2 + =
(7-3)

35. Basic Operations: Addition
2 –3 5 -2 + = 4

36. Basic Operations: Addition & Subtraction
2 –3 5 -2 + = 4
(2+5)

37. Basic Operations: Addition & Subtraction
2 –3 5 -2 + = 4 7

38. Basic Operations: Addition
2 –3 ? + =
CAN’T DO THIS!!!!!!
(must be the same order for addition/subtraction)

39. Your turn:
4. Write the result of the following:
2 –3 2 +

40. m x n (times) n x p = m x p m x p
Order of the “Product” of Two Matrices (2 matrices multiplied by each other):
Matrix A x Matrix B = AB
m x n (times) n x p
= m x p
m x p
Must be equal !!

41. Matrix Multiplication
A x B = ?
2 –3 ? x =
2 x x 2
2 x 2 =
equal
m x p

42. What is the dimension of the product?
2 –3 x ? =
2 x x 3
2 x 3 =
equal
m x p

43. What is the dimension of the product?
2 –3 x ? =
2 x x 2
NOTequal !!!!!!!
CAN’T DO THIS!!!!!!

44. So, how do you multiply matrices?
1 3
4 2 x =
1,1
1,2
2,1
2,2
Can you multiply?
What is order of
the answer matrix?

45. So, how do you multiply matrices?
1 3
4 2 x =
(1*2+3*1)
1,1
1,2
2,1
2,2
What is the “address”
Of this element?
1,1: (1st row, 1st column)

46. So, how do you multiply matrices?
1 3
4 2 x = 5
1,1
1,2
2,1
2,2

47. So, how do you multiply matrices?
1 3
4 2 x = 5
(1*-1+3*-2)
1,1
1,2
2,1
2,2

48. So, how do you multiply matrices?
1 3
4 2 x = 5 -7
1,1
1,2
2,1
2,2

49. So, how do you multiply matrices?
1 3
4 2 x = 5 -7
1,1
1,2
(4*2+2*1)
2,1
2,2

50. So, how do you multiply matrices?
1 3
4 2 x = 5 -7
1,1
1,2 10
2,1
2,2

51. So, how do you multiply matrices?
1 3
4 2 x = 5 -7
1,1
1,2
(4*-1+2*-2) 10
2,1
2,2

52. So, how do you multiply matrices?
1 3
4 2 x = 5 -7
1,1
1,2 -8 10
2,1
2,2

53. Your turn:
5. Write the product of the
two matrices.
2 –3 x ? =
2 x x 3
2 x 3 =
equal
m x p

54. Is Matrix Multiplication commutative?
B = AB -7
1 3
4 2 = x
Your Turn:
1 3
4 2 x
= ?
7. Write the product of the
two matrices.
8. Does AB = BA ?
B * A = AB

55. Matrix Multiplication using the Ti-84
2nd Matrix
Edit, Enter order
Enter elements
2nd matrix
Edit (scroll to “B” matrix
and enter elements
Clear screen (2nd “quit”)
2nd matrix, scroll to desired
matrix and enter
Enter operation desired “*”
2nd matrix, scroll to other matrix and enter, then enter again.

56. The Zero Matrix 0 0 0 Can be of any order, but every element in
Can be of any order, but every element in
the matrix is a zero.

57. Additive Inverse 4 0 2 7 -3 9 -4 0 -2 -7 3 -9 If “A” =
4 0
2 7
-3 9
“B” =
-4 0
-2 -7
3 -9
What is matrix B if matrix B is the additive inverse of matrix A?

58. 2 –3 2 7 2 4 Your turn: 6. Write the additive inverse matrix
of the following matrix
2 –3 2

59. Identity Matrix: The matrix version of the number ‘1’.

60. Your turn: 2 1 3 4 2 1 0 0 1 1 3 4 7. Multiply the following: x =
1 3
4 2 1 3 x =
1,1
1,2 2 4
2,1
2,2
The identity property of multiplication:
multiply any number by “1” and the result will be
the original number.
The identity property of multiplication:
multiply a matrix by a suitable sized identity matrix
and the result will be the original matrix.

61. Multiplicative Inverse::
w x
y z
2 –3 x =
We’ll learn next time how to find the
(mulitiplicative) inverse matrix.

62. Your Turn:

63. How would you find the values of the variables?
-2 x a 7
y b =

64. An Example: Sales of DVD Racks.
Walnut
Pine
Cherry
Last Month
125 small 100 large
278 small 251 large
225 small 270 large
This Month
95 small large
316 small 215 large
205 small 300 large
What is the average monthly DVD rack sales for the 2 month period?

65. Organize Data into a Matrix
Month A Month B
Walnut Cherry Pine Walnut cherry Pine Small
Large

66. Organize Data into a Matrix
Last Month (A) This Month (B)
small large small large
Walnut
Pine
Cherry

67. Add the matrices together and multiply by 1/2
Average monthly sales =
= ½ (A + B)
= ½

68. Add the matrices together and multiply by 1/2
Average monthly sales = ½ (A + B)
= ½

69. Add the matrices together and multiply by 1/2
Average monthly sales = ½ (A + B)
small large
Walnut
Pine =
Cherry