1. 27. Determinants and Inverses

2. Every square matrix has a whole number quantity called a determinant
The notation for the Determinant is
detA or |A|

3. Why they are important Used to find inverse of matrix
Used to find area of region
Used in decoding/encoding messages
Used to help find equation of line

4. Determinant of a 2x2
= ad - cb

5. Evaluate the determinant
=1(7)
- 4(2)
= 7 - 8
= -1

6. Evaluate the determinant
=7(3)
— 2(2) =
= 17

7. Steps for finding determinant of a 3 x 3 matrix
Step 1: recopy the first two columns.
Step 2: multiply down the diagonals and add the products
Step 3: multiply up diagonals and add the products
Step 4: Subtract the up diagonal from the down diagonal
(down – up)

8. = -73 Evaluate -3 -2 0 1 2 = (2*0*6 + -3*3*1 + 4*-2*2) _ (1*0*4 =
(2*0*6 +
-3*3*1 +
4*-2*2) _
(1*0*4
+ 2*3*2
+ 6*-2*-3)
Step 1: recopy the first two columns.
= ( ) – ( )
Step 2: multiply the
down diagonals and
add the products. =
Step 3: multiply the
up diagonals and
add the products
NOTE: You subtract the up diagonal from the down diagonal
= -73

9. You try!!!
det
= -89

10. Determinants on calculator

11. Identity In other words, 5 * __= 5?
What is the multiplicative identity for the real numbers?
In other words, 5 * __= 5?
The identity for multiplication is 1 because anything multiplied by 1 will be itself.

12. a * a-1= 1 In other words, 5 * ___=1?
What do we multiply by to get the identity?
In other words, 5 * ___=1?
a * a-1= 1
Any number multiplied by its inverse will be the identity.

13. A * A-1= I A-1 *A = I Identity Matrix
Any matrix multiplied by its inverse will be the identity matrix.
A * A-1= I
A-1 *A = I
3x3 Identity Matrix
2x2 Identity Matrix

14. Identity Matrix
Just like 5*1 = 5…
AI= A

15. Ex. 1 Determine whether A and B are inverses.
To find out if matrices are inverses:
Find out if AB = I
If so, then yes they’re inverses.
If not, then no they’re not inverses.
Ex. 1 Determine whether A and B are inverses.
YES

16. You try! Determine whether A and B are inverses.
YES

17. The Inverse of a 2x2 Matrix
As long as the determinant does not equal 0
If the determinant=0, then the matrix has no inverse!!!!

18. Ex. 4 Find A-1, if it exists.
A-1=

19. Ex. 5 You try!! Find A-1, if it exists.

20. Ex. 6 Find A-1, if it exists.
Does not exist, because it’s not square.

21. Inverses on calculator

22. Ticket out the Door What is a determinant and why does it matter?
Answer the essential question:
What is a determinant and
why does it matter?