The document discusses determinants and inverses of square matrices, specifically 2x2 and 3x3 matrices. It explains how to calculate the determinant, the significance of the identity matrix in matrix multiplication, and the procedure for finding inverse matrices. Additionally, it illustrates the application of matrices in solving systems of equations and provides a real-life investment problem to demonstrate matrix equations.

1. Determinants

2. Matrices
• A matrix is an array of
numbers that are
arranged in rows and
columns.
• A matrix is “square” if it
has the same number of
rows as columns.
• We will consider only
2x2 and 3x3 square
matrices
0
0
-½
-½
3
3
1
1
11
11
180
180
4
4
-¾
-¾
0
0
2
2
¼
¼
8
8
-3
-3

3. Determinants
• Every square matrix has
a determinant.
• The determinant of a
matrix is a number.
• We will consider the
determinants only of
2x2 and 3x3 matrices.
1
1 3
3
-½
-½ 0
0
-3
-3 8
8 ¼
¼
2
2 0
0 -¾
-¾
4
4 180
180 11
11
Note the difference in the matrix
and the determinant of the
matrix!

4. Why do we need the determinant
• It is used to help us
calculate the inverse
of a matrix and it is
used when finding
the area of a triangle

5. 4
5
2
3

Notice the different symbol:
the straight lines tell you to
find the determinant!!
(3 * 4) - (-5 * 2)
12 - (-10)
22
=
4
5
2
3

Finding Determinants of Matrices
=
=

6. 2
4
1
5
2
1
3
0
2


2
1
-1
0
-2
4
= [(2)(-2)(2) + (0)(5)(-1) + (3)(1)(4)]
[(3)(-2)(-1) + (2)(5)(4) + (0)(1)(2)]
[-8 + 0 +12]
-
- [6 + 40 + 0]
4 – 6 - 40
Finding Determinants of Matrices
=
= = -42

7. 





1
0
0
1
Identity matrix: Square matrix with 1’s on the diagonal
and zeros everywhere else
2 x 2 identity matrix










1
0
0
0
1
0
0
0
1
3 x 3 identity matrix
The identity matrix is to matrix multiplication as
___ is to regular multiplication!!!!
1
Using matrix equations

8. Multiply:






1
0
0
1





 
4
3
2
5
= 




 
4
3
2
5






1
0
0
1





 
4
3
2
5
= 




 
4
3
2
5
So, the identity matrix multiplied by any matrix
lets the “any” matrix keep its identity!
Mathematically, IA = A and AI = A !!

9. Inverse Matrix:
Using matrix equations
2 x 2






d
c
b
a
In words:
•Take the original matrix.
•Switch a and d.
•Change the signs of b and c.
•Multiply the new matrix by 1 over the determinant of the original matrix.








 a
c
b
d
bc
ad
1

 1
A

A

10. 




 



 2
4
4
10
)
4
)(
4
(
)
10
)(
2
(
1





 

 2
4
4
10
4
1
=












2
1
1
1
2
5
Using matrix equations
Example: Find the inverse of A.







 10
4
4
2

A

 1
A

 1
A

11. Find the inverse matrix.








2
5
3
8
Det A = 8(2) – (-5)(-3) = 16 – 15 = 1
Matrix A
Inverse =










det
1 Matrix
Reloaded






8
5
3
2
1
1
= = 





8
5
3
2

12. What happens when you multiply a matrix by its inverse?
1st
: What happens when you multiply a number by its inverse?
7
1
7 
A & B are inverses. Multiply them.






8
5
3
2
=








2
5
3
8






1
0
0
1
So, AA-1
= I

13. Why do we need to know all this? To Solve Problems!
Solve for Matrix X.
=








2
5
3
8
X 







1
3
1
4
We need to “undo” the coefficient matrix. Multiply it by its INVERSE!






8
5
3
2
=








2
5
3
8
X 





8
5
3
2








1
3
1
4






1
0
0
1
X = 







3
4
1
1
X =








3
4
1
1

14. You can take a system of equations and write it with
matrices!!!
3x + 2y = 11
2x + y = 8
becomes 





1
2
2
3






y
x
= 





8
11
Coefficient
matrix
Variable
matrix
Answer matrix
Using matrix equations

15. Let A be the coefficient matrix.
Multiply both sides of the equation by the inverse of A.










































8
11
8
11
8
11
1
1
1
A
y
x
A
y
x
A
A
y
x
A






1
2
2
3 -1
= 







 3
2
2
1
1
1
= 







3
2
2
1








3
2
2
1






1
2
2
3






y
x
= 







3
2
2
1






8
11






1
0
0
1






y
x
= 





 2
5






y
x
= 





 2
5
Using matrix equations






1
2
2
3






y
x
= 





8
11
Example: Solve for x and y .

 1
A

16. Wow!!!!
3x + 2y = 11
2x + y = 8
x = 5; y = -2
3(5) + 2(-2) = 11
2(5) + (-2) = 8
It works!!!!
Using matrix equations
Check:

17. You Try…
Solve:
4x + 6y = 14
2x – 5y = -9
(1/2, 2)

18. You Try…
Solve:
2x + 3y + z = -1
3x + 3y + z = 1
2x + 4y + z = -2
(2, -1, -2)

19. Real Life Example:
You have $10,000 to invest. You want to invest the money
in a stock mutual fund, a bond mutual fund, and a money
market fund. The expected annual returns for these funds
are given in the table.
You want your investment to obtain an overall annual
return of 8%. A financial planner recommends that you
invest the same amount in stocks as in bonds and the money
market combined. How much should you invest in each
fund?

21. To isolate the variable matrix, RIGHT multiply by the inverse of A
1 1
A AX A B
 

1
X A B


Solution: ( 5000, 2500, 2500)