1. MATRICES
Adapted from presentation found on the internet. Thank you to the creator of the original presentation!

2. What is a Matrix?
A matrix is a set of elements, organized into rows and columns
rows
columns

3. Plural form is matrices.
Matrix Essentials
Plural form is matrices.
The “things” in a matrix are called elements or members.

4. Matrix Essentials (continued)
The dimensions of a matrix are determined by rows x columns.
C3x4 means that matrix C has 3 rows and 4 columns
E5x2 means that matrix E has 5 rows and 2 columns
C3x4 is different from C3,4 which means the element in the 3rd row and 4th column of matrix C.

5. Element Identification
E3x5 = [ ]
Identify the requested elements.
What is E3,1? == -9
What is E1,3? == 4
What is E1,2? == -2
What is E2,1? == 3
What is E2,5? == 23
What is E5,2? == does not exist. There are only 3 rows.
What is E2,4? == 7
What is E4,2? == does not exist. There are only 3 rows.

6. Row Matrix
Row matrix
matrix having only 1 row
EX: B1x3 = [ ]

7. Column matrix Column Matrix matrix having only 1 column EX: D2x1 = [ ]3 8

8. Square Matrix
Matrix in which the number of rows is the same as the number of columns.
There are three things that only exist with a square matrix (more on these later):
determinant
identity matrix
inverse matrix

9. Two things must occur to have equivalent matrices:
They have the same dimensions.
Corresponding elements have the same value.

10. Addition and Subtraction
Addition and subtraction can only be done if the dimensions of the two matrices match.
A 3x4 matrix can be added to another 3x4 matrix, BUT …
A 2x6 matrix cannot be added to a 2x4 matrix. (# of columns does not match)
A 3x2 matrix cannot be added to a 2x3 matrix. (Neither dimension matches. Don’t get tripped up by this one.)

11. Addition and Subtraction (cont.)
Just add corresponding elements
Just subtract corresponding elements

12. Scalar Multiplication
Scalar multiplication involves nothing more than multiplying a number or variable outside a matrix by every term inside the matrix.
3[ ] =
4 3 -2
7 5 9
[ ]

13. Matrix Multiplication
Two matrices can be multiplied only if the number of columns from the 1st matrix equals the number of rows from the 2nd matrix.
Can the following matrices be multiplied?
A2x3 and B3x4?
C3x1 and D3x1?
E3x2 and F2x3?
G2x2 and H2x2?

14. Size of Product Matrix
The product has the same number of rows as the first matrix and the same number of columns as the second matrix.
Size is based on the outer dimensions.

15. Matrix Multiplication with Square Matrices
Multiply each element in a row by the corresponding element in the column of the second matrix.
It is a lot easier to show matrix multiplication through examples worked out on the board, so pay attention to the examples given in class.

16. Commutativity of Matrix Multiplication
Is AB = BA? Maybe, but maybe not!
Multiplication may be possible but resulting in different elements in the answer.
Multiplication may be possible but resulting in different size product matrix
A3x1 ●B1x3 = P3x3 BUT . . .
B1x3 ● A3x1 = P1x1

17. Commutativity (continued)
A third reason for matrix multiplication not to be commutative is that the multiplication may be possible in one direction but not in the other.
A2x4 ● B4x5 = P2x5 BUT …
B4x5 ● A2x4 cannot be multiplied.

18. Determinant of a Matrix
Used for inversion
If det(A) = 0, then A has no inverse.
Square matrix always has determinant.
Symbol is
Inverse matrices can be used in solving a system of linear equations.

19. Determinant of a 3x3 Matrix
Sum from left to right
Subtract from right to left
Note: N! terms
There is a less jumbled way of doing this. Pay attention to the examples worked out on the board.
Mr. Parker’s note: We will do ours in a less jumbled, more organized way.

20. Determinant of a 3x3 Matrix
Rewrite first two columns. Find “down” total. Find “up” total.
Det = DOWN – UP

21. Determinants and Area of Triangles in the Coordinate Plane
If you know the coordinates of the vertices of a triangle in the coordinate plane, you can use the absolute value of a 3x3 determinant to find the area of the triangle.
Coordinates are
(x1, y1)
(x2, y2) and
(x3, y3)
Area is | |
x1 y1 1
x2 y2 1
x3 y3 1 1 2

22. Identity Matrix Identity matrix only exists with a square matrix.
On principal diagonal the elements are 1s. Everywhere else is 0s.
Identity matrix: AI = A
Principal diagonal is where the row position and column position are the same. In other words, in an identity matrix like the one shown, I1,1 = 1, I2,2 = 1, I3,3 = 1, but I1,3 = 0 and I2,1 = 0.

23. Inverse of a Matrix
When discussing inverse, it is implied that we are talking about the multiplicative inverse.
Inverse only exists with square matrices.
If the inverse exists, then: AA-1 = I
2x2 inverse can be done easily by hand. 3x3 inverse is better found by using technology.

24. Inverse of a 2x2 Matrix
* As we said before, the inverse of A exists only as long as det A ≠ 0. or

25. Inverse of a Matrix Append the identity matrix to A
Subtract multiples of the other rows from the first row to reduce the diagonal element to 1
Transform the identity matrix as you go
When the original matrix is the identity, the identity has become the inverse!