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Multiplication of Matrices
EQ: How do you multiply two matrices?
Notes:
Find the Product of two Matrices
Restrictions on Multiplying:
•The number of columns of the first matrix must be
equal the number of rows in the second matrix.
Ex: Can we multiply these two? Why or why
3 0 1 
not? 1 3
A

0
2


dim: 2 x 2
B

7

2
4


dim:2 x 3
Yes, because the column=row
Ex: Can we multiply these two? Why or why not?
0
3
A


5

1


B  2 3
dim: 2 x 2
dim:1 x 2
NO, because the dimensions are NOT the same!
NOTES:
Find the Product of two Matrices
Now that you have determined the multiplication is defined
Follow the
Steps to Multiplying:
1. Create an ‘answer matrix’ with dimensions
using row count of first matrix and column
count of second matrix.
Ex: 1
1 3
A

0
2


dim: 2 x 2
 2 3
 __
B
Answer ( A  B)  

  1 4
 __
dim:2 x 2
Ex: 2
A  3  1
dim: 1 x 2
__ 
__ 
dim: 2 x 2
4  1
B
Answer ( A  B)  ___

2 0 
dim:2 x 2
dim: 1 x 2
___ 
Find the product of two matrices
Notes:
Steps continued:
2. Multiply the numbers in the first row of A by the
numbers in the first column of B, multiplying
first-x-first, second-x-second, etc. until all entries
in the first row are completed;
3. Add the products, and put the result in the first
row, first column of AB.
1
3
4
–2
5 –7
9
6
=
1(5) + 4(9) _______
________ _______
EQ: How do you multiply two matrices?
Find the product of two matrices
Notes:
Steps continued:
4. Repeat steps 2 & 3, this time with the numbers in
the first row of A the numbers in the second
column of B, multiplying first-x-first, second-xsecond, etc. until all entries in the first row are
multiplied.
5. Add the products, and put the result in the first
row, first column of AB.
1
3
4
–2
5 –7
=
9
6
1(5) + 4(9)
1( – 7) + 4(6)
6. Repeat until the last column of B has been
multiplied against that first row of A .
EQ: How do you multiply two matrices?
Find the product of two matrices
Notes:
Steps continued
7. Move to the second row of A and multiply with
the numbers in the first column of B, add the
products, and put the result in the second row,
first column of AB.
1
3
4
–2
5 –7
=
9
6
1(5) + 4(9)
3(5) + (– 2)(9)
1( – 7) + 4(6)
8. Repeat until the entries in all rows are multiplied
by the entries in each column.
1(7)  4(6)   41 17 
 1(5)  4(9)
3(5)  (2)(9) 3(7)  (2)(6)   3  33

 

GUIDED PRACTICE
Number 1
–3
1. Find A∙B if A =
1
SOLUTION
3
–2
and B =
2x2
1 1 0
–3 2 –1
2x3
Because the column count of A, equals the row
count of the B, the product AB is defined and is a
2 x 2 matrix.
EQ: How do you multiply two matrices?
GUIDED PRACTICE
Number 1
Solution (continued)
Repeat steps 2- 6 for the first row of A by the each
column of B, add the products, and put the results in
the first row of the answer matrix.
–3 3
1 –2
2x2
–3(1)+
3 (–3) _________
–3(1)+ 3 (2) _________
–3(0)+ 3 (1)
________
1
1
0
.
= 1(1)+(3) (–3) ________
1(1)+(- 3) (2) ________
1(0)+(- 3) (1)
_______
–3 2 1
2x3 
Repeat steps 2- 6 for the second row of A by the
each column of B, add the products, and put the
results in the first row of the answer matrix.
Simplify!
3 
 12 3
 10  5  3 


GUIDED PRACTICE
Number 2
Second – you try!
2  1 
 1 3 


3
4
  2 0 


 0  2


NO, dimensions are not matching.
Undefined.
EQ: How do you multiply two matrices?
GUIDED PRACTICE
Number 3
Once again – you try!
1 0  1 4 
3 2 0  2  1 0  2 7 


NO, dimensions are not matching.
Undefined.
GUIDED PRACTICE
Number 4
Once again – you try!
1  2 1
4 7
2

0  1 3

5 3  1
0  1 
1  0 


5   2
  
1  7 
Yes, dimensions are matching. 1
7
 
 29 
 
14 
Homework Check
Adding/Subtracting Matrices
Clear desks, calculators out!