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Chapter Seven
Linear Systems and Matrices
7.5 – Operations with Matrices
• Equality of Matrices
• Matrix Addition and Scalar Multiplication
• Matrix Multiplication
• Applications
7.5 – Equality of Matrices
•
•
A matrix is equal to another if the
dimensions are the same and the entries
are all equivilant.
Ex: Solve for x and y:
6 
 2 x  y 6  5

 1


10 1  x  2 y 

x4
y  3
7.5 – Addition of Matrices Example:
•
Matrix addition is a piece-wise addition
and therefore the dimensions need to be
exactly the same.
Ex: Find A+B
 1 8 
0 4 
A
B


 6 2
5 3
 1 12 
11 1


•To add matrices:
•1. Check to see if the matrices have the same order.
•2. Add corresponding entries.
•Example: Find the sums A + B and B + C.
1 5 
2 0 6
3 3 0 


A   2 1 B  
C



1
0

3
3
2
4




 0 6 
•A has order 3 2 and B has order 2  3. So they cannot
be added. •C has order 2 3 and can be added to B.
 2 0 6  3 3 0  5 3 6
BC  





1 0 3 3 2 4  2 2 1 
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•To subtract matrices:
•1. Check to see if the matrices have the same order.
•2. Subtract corresponding entries.
•Example: Find the differences A – B and B – C.
3 7
 2 1
 1 5 1 
A
B
C 



2
1
4

5
2
1
6






•A and B are both of order 2 2 and can be subtracted.
 3 7  2 1  1 8
A B  





 2 1  4 5  2 6
•Since B is of order 2  2 and C is of order 3  2,
they cannot be subtracted.
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•If A is an m n matrix and C is a scalar , then the
m n matrix CA is the scalar multiple of A by C.
 2 5 1
3 4 0 
•Example: Find 2A and –3A for A =
.
 2 7 2 
 2(2) 2(5) 2(1)   4
2 A   2(3) 2(4) 2(0)   6
 2(2) 2(7) 2(2)   4
 3(2) 3(5) 3(1)  6
3 A   3(3) 3(4) 3(0)   9
 3(2) 3(7) 3(2)   6
9
10 2 
8 0 
14 4 
15 3 
12 0 
21 6
•Example: Calculate the value of 3A – 2B + C with
2
2
 2 1
5
5
A   3
5  B   1 0  and C   1 0 
 4  2 
 3  1 
 3  1 
 2 1
5 2  5 2 
3 A  2 B  C  3  3 5   2 1 0   1 0 
 4 2 
 3 1  3 1
 6 3  10 4  5 2  1  5
  9 15    2 0   1 0    8 15 
12 6   6 2   3 1  9  5 
10
7.5 – Matrix Equations
With matrix equations, the variable you are
solving for is a matrix.
Ex: Solve the matrix equation 3X+A=B given
matrices A and B:
1 2
 3 4
A
B


0 3 
 2 1
 4
 3

 2
 3

2 

2

3 
7.5 – Matrix Multiplication
•
2 matrices have a product if the # of
columns of the left matrix = the # of rows
of the right matrix. Multiply across on the
left, down on the right.
Ex: Find the product AB, then try BA.
 1 3 
 3 2


A   4 2 B  

4 1 

 5 0 
 9 1 
 4 6 


 15 10
7.5 – The Identity Matrix
This is a square matrix in which all of the
diagonal entries are ones and all of the
off-diagonal entries are zero.
Ex: Multiply matrix A by the identity matrix I.
3 2 5
1 0 0 




A   1 0 4  I  0 1 0 
 1 2 3
0 0 1 
3 2 5
1 0 4


 1 2 3
7.5 – Application
• Find the equation of the parabola
2
y  ax  bx  cthat passes through the points.
 5,6 , 1,0 ,  2, 20
y  3x  11x  14
2
7.5 – Application
• An inheritance of $20,000 is divided
among 3 investments yielding $1780 in
interest per year. The interest rates for the
three investments are 7%, 9%, and 11%.
Find the amount of each if the amount
invested at 7% was $2000 less than half of
the total investment.
$8000at 7%, $5000at 9%, $7000at11%
Homework
• 7.4 pg. 501: 71,73
• 7.5 pg.514 1-7odd, 15, 23-29 odd, 65,67