Determinants, Inverse Matrices & Solving
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
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 !!
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
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)
You Try…
Solve: 2x + 3y + z = -13x + 3y + z = 12x + 4y + z = -2
(2, -1, -2)
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?
To isolate the variable matrix, RIGHT multiply by the inverse of A
1 1A AX A B 1X A B
Solution: ( 5000, 2500, 2500)
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