day 3 - Cobb Learning
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Transcript day 3 - Cobb Learning
Identity & Inverse Matrices
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.
Inverses
What does “inverse” mean to you?
What is the inverse of multiplication?
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
The multiplicative identity for
matrices is a square matrix with
ones on the main diagonal and
zeros everywhere else.
1 0
I
0 1
1 0 0
I 0 1 0
0 0 1
Identity Matrix
Just like 5*1 = 5…
AI= A
2
1
8
IA= A
3 1 0 2
17 0 1 1
8
3
17
Or
1 0 2 3 2 3
0 1 1 17 1 17
8
8
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
3x3 Identity Matrix
1 0 0
I 0 1 0
0 0 1
Ex. 1 Determine whether A and B are
inverses.
2 1
2 3
A
2
B
1
3
6
3
YES
Ex. 2 Determine whether A and B are
inverses.
4 3
5 3
B
A
7 5
7 4
NO
The Inverse of a 2x2 Matrix
a b
A
c d
-1
A =
1
A
If ad-cd=0, then the matrix
has no inverse!!!!
d b
c a
1 d b
ad bc c a
As long as ad-cb =0
Ex. 3 Find A-1, if it exists.
2 3
A
5 7
-1
A =
1 7 3
14 15 5 2
7 3
A
5 2
1
Ex. 4 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. 5 Find A-1, if it exists.
3 4
A 2 6
1 0
Does not exist,
because it’s not
square.
Now let’s learn how to
use our calculator!!!
Find the inverse!
2 3
A
5 7
2 1
A
4 0
Yes, now you can add, subtract, multiply,
and find the determinant in you calculator!!
Solving Systems using
Matrices and Inverses
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
Solving a Matrix Equation
Solve the matrix equation AX=B for
the 2x2 matrix X
4 1
8 5
X
3 1
6 3
X = A-1B
2 2
X
0 3
Ex. Solve
3 4
3 8
X
5 7
2 2
29 48
X
21 34
Solving Systems Using
Inverse Matrices
5x 2 y 3
4x 2 y 4
Setting Up the Matrices
• Matrix A will be the coefficients of
the system
• Matrix X will be the variables
• Matrix B will be constants (what
the system of equations are equal
to)
Matrix Equation
A linear system can be written as a
matrix equation AX=B
5 4 x 8
1 2 y 6
Constant
Coefficient
matrix
Variable
matrix
matrix
5x 4 y 8
1x 2 y 6
Example 1
5x 4 y 8
1x 2 y 6
5 4 x 8
1 2 y 6
Example 2:
Use matrices to solve the linear system
5x 2 y 3
4x 2 y 4
5 2 x
4 2 y
3
4
Type in [A]-1 [B]
Find the inverse
1 1
3
x
5
y
2
4
2
(-1, 4)
Example 3:
Use matrices to solve the linear system
4x 2 y 8
x 2 y 12
4 2 x 8
1 2 y 12
Type in [A]-1 [B]
Find the inverse
x .2 .2 8
y .1 .4 12
(4, 4)
Example 4:
Use matrices to solve the linear system
x y 2z 3
2 x y 3z 4
4 x 3 y z 18
2 x 3
1 1
2 1 3 y 4
4 3 1 z 18
Type in [A]-1 [B]
(-2, 3, 1)
Example 5:
Use matrices to solve the linear system
2x z 2
5x y z 5
x 2 y 2z 0
2 0 1 x 2
5 1 1 y 5
1 2 2 z 0
Type in [A]-1 [B]
(2, 3, -2)
Let’s apply this…
You have $18 to spend for lunch during a 5
day school week. It costs you $1.50 to make
lunch at home and $5 to buy lunch. How
many times each week do you make a lunch
at home?
x y 5
1.5 x 5 y 18
(2, 3)
1 1 x 5
1.5 5 y 18
Type in [A]-1 [B]
You make lunch at home 2 times a week.
A word problem…!!
• A small corporation borrowed $1,500,000 to
expand its product line. Some of the money was
borrowed at 8%, some at 9% and some at 12%.
How much was borrowed at each rate if the
annual interest was $133,000 and the amount
borrowed at 8% was 4 times the amount
borrowed at 12%?
$800,000 at 8%
$500,000 at 9%
$200,000 at 12%
Homework