Transcript Warm-Up - s3.amazonaws.com

Bellwork
1) Multiply.
 3 3 
0
 3  2   1

  2  1

 0  1 
2) Multiply.
 2 9 6


2 3  4   3 3 4


 2  2 1
3) Find the
determinant.
 12 2
 5 8 


4) Find the determinant
 15 4  10
 10 0 6 


 8 2  14
Answers
 9  3
7

2


2
1 
13
3.)106
4.)  12
-1 - 4
3.8A Use Inverse
Matrices to solve linear
systems
Algebra II
Identity
What does “identity” mean to you?
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.
What do we multiply by to get the identity?
In other words, 5 * ___=1?
a *
-1
a =
1
Any number multiplied by its inverse
will be the identity.
Identity Matrix
Any matrix multiplied by its inverse
will be the identity matrix.
A *
-1
A
-1
A =
I
*A = I
2x2 Identity Matrix
1 0
I 

0 1
Identity Matrix
Just like 5*1 = 5…
AI= A
2
1
 8
IA= A
 3 1 0  2




1


17 
 0 1  8
 3

17 

Or
1 0  2  3  2  3
0 1   1 17    1 17 
  8

  8

Determine whether A and B are
inverses.
1

3 1 
 1  2
A
B


3
4
2


 2

2 

Find out if AB = I
If so, then yes they’re inverses.
If not, then no they’re not inverses.
Use your calculator.
YES
The Inverse of a 2x2 Matrix
a b 
A

c d 
-1
A =
1
A
 d  b
1  d  b
 c a   ad  bc  c a 




As long as ad-cb =0
If ad-cb=0, then the matrix has no
inverse!!!!
Ex. 1 Find A-1, if it exists.
2 1
A

 4 0
-1
A =
1 

0
0

1




1
4




1
 4  4 2  1  
2

Ex. 2 Find A-1, if it exists.
3 4 


A   2  6
1 0 
Does not exist,
because it’s not
square.
Solving Matrix Equations
Suppose ax = b
How do you solve for x?
We cannot divide
matrices, but we
can multiply by the
inverse.
A-1 AX =A-1 B
IX = A-1B
X = A-1B
Ex. 3 Solving a Matrix Equation
Solve the matrix equation AX=B for
the 2x2 matrix X
 4  1
 8  5
X

 3 1 
 6 3 




 4  1
 8  5
 3 1  X   6 3 




Step 1: find A-1
A-1=
1 1 1 1 1
 1


4  3 3 4 3 4
Step 2: Multiply both sides of the equation
By A-1 on the left
1 1  4  1
1 1  8  5
3 4  3 1  X  3 4  6 3 






1 1  4  1
1 1  8  5
3 4  3 1  X  3 4  6 3 






53 
 4  3 1  1 
 86
12  12  3  4 X  24  24  15  12




1 0
 2  2
0 1  X   0  3 




You can check the
Solution by mult
AX to see if you get B
 2  2
X 

0  3
Assignment
Inverses
What does “inverse” mean to you?
What is the inverse of multiplication?
Determine whether A and B are inverses.
 5  3
A

 7 4 
 4 3
B

  7 5
NO