1. [MATRICES ]

3. Matrix (plural: matrices)
a rectangular array of numbers written within brackets
Represented with a capital letter and classify by its dimension
Dimensions of a Matrix/Order of a Matrix
determine by the number of horizontal rows and the number of vertical columns
Matrix Element
each number in a matrix

4. Writing the dimensions of a matrix.
4 columns A=
3 rows
Matrix A is a 3 x 4 matrix.

5. Write the dimensions or order of each matrix.
6 5
-4 1/3 -3
3 X 3
1 X 3
10 0
1 -5
5 0
3 X 2
2 X 3

6. Identifying a Matrix Element
aij denotes the element of the matrix A on the ith row and jth column.
Example:
Identify element a13 in Matrix A.
Answer: a13 means the element in row 1, column 3.
a13 = 1

7. Identify each matrix element

8. Adding and Subtracting Matrices
to add or subtract matrices A and B with the same dimensions, add or subtract the corresponding elements
***Note: you can only add or subtract matrices with the same dimensions.

9. Properties: Matrix Addition
If A, B, and C are m x n matrices, then
a. Closure Property
A + B is an m x n matrix
Commutative Property
A + B = B + A
Associative Property for Addition
(A + B) + C = A + (B + C)
Additive Identity Property
There exist a unique m x n matrix O such that
O + A = A + O = A
Additive Inverse Property
For each A, there exists a unique opposite –A.
A + (-A) = O

10. Find the sum or difference of each matrix.1. 1.
-2 0
3 -5 7
9 -3 + = 2.
-12 24
-3 5
-1 10
-3 1
2 -4
-1 5 +
-1 1 = 3.
-9 7 -
-12 7 = 4.
-3 5
-1 10
-3 1
2 -4 -
0 4 =

11. Identify whether the two matrices are additive inverse or not.1. 5
0 -2
-14 -5
0 2 ,
Yes.
Find the additive inverse of the given matrix. 1.
-10 5
0 -2 3
=>

12. Solving Matrix Equations
an equation in which the variable is a matrix
Equal Matrices
matrices with the same dimensions and with equal corresponding elements

13. Solving a Matrix Equation
Solve for the matrix X.
Solution: 1
0 1
8 9 =
X - 1
0 1
8 9 =
X - 1
0 1
8 9 +
X =
1 2
11 11
X =

14. Do these… = - X = Solve for Matrix X. 1. X + 2. -1 0 2 5 10 7 -4 4 7
-1 0
10 7
-4 4 =
X + 7
-6 -1
Answer: X =
- X =
Answer: X =

15. Determining Equal Matrices
Determine whether the two matrices in each pair are equal. 1. 2. 4 6 8 ,
8/2 18/3 16/2
No, because they do not have the same dimensions.
5 0
-8/4 6 – 3
15/ ,
Yes, because they have the same dimensions and the corresponding elements are equal.

16. Finding Unknown Matrix Elements
Solve the equation for x and y. x
3 -y
38 -5
3 4y – 10 =
Solution:
x + 8 = 38
x = 30
-y = 4y – 10
-5y = -10
y = 2

17. Do these… , = Solve each unknown variable in each equation. 1. 2.
3x 4
-9 x + y ,
x = -3, y = 7 4
8 12
4x – t + 5x
4x t +1.5x =
x = 2; t = 3/5