PRECALCULUS 2 Determinants, Inverse Matrices & Solving
Mar 26, 2015
PRECALCULUS 2
Determinants, Inverse Matrices & Solving
December
• DO NOW:
TAKE OUT PAPER OR A NOTEBOOK!!!!
CW: INVERSE POWER POINT!!!
HW: INVERSE wksts.
• SWBAT:
• Identify the inverse of a 2x2 matrix
• Identify the inverse of a 3 x 3 matrix
45
23
Notice the different symbol:
the straight lines tell you to
find the determinant!!
(3 * 4) - (-5 * 2)
12 - (-10) 22
=45
23
Finding Determinants of Matrices
=
=
241
521
302
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
Inverse Matrix:
Using matrix equations
2 x 2
dc
ba
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.
ac
bd
bcad
1 1A
A
24
410
)4)(4()10)(2(1
24
410
41
=
21
1
125
Using matrix equations
Example: Find the inverse of A.
104
42A
1A
1A
Find the inverse matrix.
25
38
Det A = 8(2) – (-5)(-3) = 16 – 15 = 1
Matrix A
Inverse =
det
1 MatrixReloaded
85
3211
= =
85
32
10
01
Identity matrix: Square matrix with 1’s on the diagonal and zeros everywhere else
2 x 2 identity matrix
100
010
001
3 x 3 identity matrix
The identity matrix is to matrix multiplication as ___ is to regular multiplication!!!!1
Using matrix equations
Multiply:
10
01
43
25=
43
25
10
01
43
25=
43
25
So, the identity matrix multiplied by any matrix lets the “any” matrix keep its identity!
Mathematically, IA = A and AI = A !!
What happens when you multiply a matrix by its inverse?
1st: What happens when you multiply a number by its inverse?71
7
A & B are inverses. Multiply them.
85
32=
25
38
10
01
So, AA-1 = I
Why do we need to know all this?To Solve Problems!Solve for Matrix X.
=
25
38X
13
14
We need to “undo” the coefficient matrix. Multiply it by its INVERSE!
85
32=
25
38X
85
32
13
14
10
01X =
34
11
X =
34
11
You can take a system of equations and write it with
matrices!!!
3x + 2y = 11
2x + y = 8becomes
12
23
y
x=
8
11
Coefficient
matrix
Variable
matrix
Answer matrix
Using matrix equations
Let A be the coefficient matrix.
Multiply both sides of the equation by the inverse of A.
8
11
8
11
8
11
1
11
Ay
x
Ay
xAA
y
xA
12
23 -1=
32
21
11
=
32
21
32
21
12
23
y
x=
32
21
8
11
10
01
y
x=
2
5
y
x=
2
5
Using matrix equations
12
23
y
x=
8
11Example: Solve for x and y .
1A
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:
You Try…
Solve:
4x + 6y = 142x – 5y = -9
(1/2, 2)