Download Ch. 9-5

Section 9.5
The Algebra of Matrices
Objectives:
• To understand matrix addition and
subtraction.
• To understand scalar and matrix
multiplication.
Equality of Matrices
Two matrices are equal if they have
the same entries in the same positions.
Equal Matrices
 4 22 e 0   2 4 1

  1 2

0

0.5 1 1  1  2 2
Unequal Matrices
 1 2
3 4    1 3 5 

 2 4 6 

5 6  
Ex 1. Determine if the matrices are equal.

 2 7 0  2
1 5 0   

 30

14 
 
2

5 
Addition, Subtraction, and Scalar Multiplication
• Two matrices can be added or subtracted
if they have the same dimension.
– Otherwise, their sum or difference is undefined.
– We add or subtract the matrices by adding
or subtracting corresponding entries.
Scalar Multiplication
• To multiply a matrix by a number,
we multiply every element of the matrix
by that number.
Ex 2. Carry out each indicated operation, or explain
why it cannot be performed.
(a) A + B (b) C – D (c) C + A (d) 5A
2 3 


A  0 5 
7  21 
7 3 0 
C

0 1 5 
1

B   3
 2
6
D
8
0

1
2
0 6 

1 9
Class Work
Determine if the matrices are equal.
1
1. 
5
3
2
5

 8 log 4 4

2  
.5
 4 8
 7 5 




A   1 12  B   4 20
 7 9 
 8 11 
Find.
2.A + B
3.2A
4.3A - B
HW p684 1-6 all, 17-23 odd
Class Work
1. Write the system as an augmented matrix
and then solve.
 x  9 y  7 z  41
8 x  2 y  6 z  70
3 x  5 y  4 z  27
Perform the indicated operation or explain why
it is not possible.
 9 19 11
 3 6 6
P
Q


4
2

13

18

11
2




2. Q – P
3. P + R
4. 6R
 3 6 
R   4 7 
11 5
Matrix Multiplication
If A is an m x n matrix and B is an n x k matrix
then AB has dimension m x k.
That is, two matrices can be multiplied if and
only if the number of columns in A matches
the number of rows in B.
Ex 3. Find AB.
 1 3
 1 5 2
A
B


 1 0
 0 4 7
Ex 4. Find CD.
4 1
 1 2


C   3 0  D  

3 1

 2 1
Class Work
5. Find DC.
4 1
 1 2


C   3 0  D  

3 1

 2 1
 5 7
1 2 
A
B




 3 0 
9 1
6. Find AB
7. Find BA.
Writing a system as a Matrix Equation
A matrix equation is in the form AX = C where A
is a coefficent matrix, X is a variable matrix,
and C is the constant matrix.
Ex 5. Write the system as a matrix equation.
 x  y  3z  4

 x  2y  2z  10
3 x  y  5z  14

Class Work
8. Write the system as a matrix equation.
 x y z  6

 4 x  6y  8z  20
2x  5y  3z  19

HW p685 18, 20, 22, 25, 27,
28, 31, 35, 43, 46