Download Matrices - Thomasville High School

Matrix (plural: matrices)
• Matrix
– 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
Writing the dimensions of a matrix.
5 0 1 2
 3 4 9 2
 3 1 4 2
Matrix A is a 3 x 4 matrix.

Write the dimensions or order of each
matrix.
1.
3.
4 6 5
-4 1/3 -3
2 -3 -7
1 0 9
1X3
3X3
2.
4.
10 0
4
5
0
1
-5
-2 0.5 17
-6.2 9
2X3
3X2
Identifying a Matrix Element
column j 
5 0 1 
A  row i 2 3 4
9 2 6 

 3 1 4 
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
Identify each matrix element
5 0 1 2
 3 4 9 2
 3 1 4 2
1.
2.
3.
4.
5.
a33
a11
a21
a34
a23
1.
2.
3.
4.
5.
a33 = 4
a11 = -5
a21 = 3
a34 = 2
a23 = -9
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.
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
b.
Commutative Property
c.
Associative Property for Addition
d.
Additive Identity Property
There exist a unique m x n matrix O such that
O+A=A+O=A
e.
Additive Inverse Property
For each A, there exists a unique opposite –A.
A + (-A) = O
A+B=B+A
(A + B) + C = A + (B + C)
Find the sum or difference of each matrix.
1. 1.
1 -2
3 -5
2. -12
-3
-1
0
7
24
5
10
3. 6 -9 7
-2 1 8
4. -3
-1
5
10
+
3 9 -3
-9 6 12
+
-3
2
-1
-
-4 3 0
6 5 10
=
10
-8
-
-3
2
=
0
-3
=
4
-6
1
-4
5
=
-15
-1
-2
1
-4
7
1
-3
19
25
1
15
-12
-4
7
-2
4
14
Identify whether the two matrices are
additive inverse or not.
1. 14
0
5
-2
,
-14
0
-5
2
Yes.
Find the additive inverse of the given
matrix.
1. -1
0
10
2
-5
-3
=> 1 -10 5
0 -2 3
Solving Matrix Equations
• Matrix Equation
– an equation in which the variable is a
matrix
• Equal Matrices
– matrices with the same dimensions and
with equal corresponding elements
Solving a Matrix Equation
• Solve for the matrix X.
0
1
1
X = 8
3
2
1
9
• Solution:
X -
1
3
1
2
=
0
8
1
9
X =
0
8
1
9
X =
1
11
2
11
+
1
3
1
2
Do these…
Solve for Matrix X.
0
1. X + -1
=
2
5
Answer:
2.
2
0
1
2
Answer:
X=
10
-4
11
-6
7
4
7
-1
-1
1
-X =
X=
-9
-15
-2
-11
11
15
3
-9
12
-7
-13
8
Determining Equal Matrices
Determine whether the two matrices in
each pair are equal.
1.
4
6
8
,
8/2
18/3
16/2
No, because they do not have the same dimensions.
2.
-2
5
3
0
,
-8/4
15/3
6–3
4-4
Yes, because they have the same dimensions and
the corresponding elements are equal.
Finding Unknown Matrix Elements
• Solve the equation for x and y.
x+8
3
-5
-y
=
38
3
Solution:
x + 8 = 38
x = 30
-y = 4y – 10
-5y = -10
y=2
-5
4y – 10
Do these…
Solve each unknown variable in each
equation.
1.
,
3x
-9
4
x+y
x = -3, y = 7
2.
2
8
4
12
=
4x – 6
4x
x = 2; t = 3/5
-10t + 5x
15t +1.5x