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Solving Using Matrices

- By Helen Chin
- Kitty Luo
- Anita La

How to find the Determinant

How to find the Inverse

-1

Example of (2 x 2) Inverse

-1

4 -2 -3 1

1 (4x1) - (2x3) -2

Extra Inverse Information

Why are inverses useful? How would you solve 5x

10? Normally you would simply divide both sides

by 5, but matrices cannot be divided. There is

another method -- multiplying each side by the

inverse. 10 times 1/5 is still 2 division and

multiplying by the inverse are the same thing. A

number times its inverse will always equal one.

You would solve matrix equations with inverses.

Solving Matrix Equations with Inverses

B

A

4 0 6 3 -1 5

2 4 0 1

x

B

inverse of A

1/2 -2 0 1

4 0 6 3 -1 5

x

Multiply each side by the inverse of A. The

inverse of A multiplied by A would cancel each

other out, leaving x inverse of A times matrix

B.

-4 2 -7 3 -1

5

x

Cramer's Rule

- ...is a way of solving for one variable instead

of solving the entire system of equations. - x - y z 2 1 -1 1
- 2x y 4z 4 2 1 4
- -x 3y - z 6 -1 3 -1

x y z

First, set up the coefficient matrix. Be sure to

put the variables' values in the same column as

the others. (So the x values would always be in

the same column, etc.)

matrix A

1 -1 1 2 1 4 -1 3 -1

Find the determinant of matrix A. In this case,

it would be -4.

Let's say you want to solve for x.

Replace the x column values with the answer

column values of the system.

x - y z 2 2x y 4z 4 -x 3y - z 6

2 -1 1 4 1 4 6 3 -1

matrix X

- Find the determinant of the new matrix. In this

case, it would be -48. You would then divide the

determinant of matrix X by the determinant of

matrix A to get the x value.

-48/-4 x is 12

Do the same thing to solve for y and z.

1 -1 2 2 1 4 -1 3 6

1 2 1 2 4 4 -1 6 -1

z

y

determinant of matrix Y is -16

determinant of matrix Z is 24

y determinant A / determinant Y

z determinant A / determinant Z

-16/-4

24/-4

y 4

z -6

Word Problems

- You inherit 60,000 from a distant relative. You

decide to invest it in three different stocks

with returns of 2, 8, and 10 respectively. You

place 5000 more in the second stock than the

first and third stock combined. You receive an

annual return of 6 of the original 60,000. How

much did you invest in each stock?

First, write 3 equations that represent this

situation.

B A C 5000 A B C 60,000 .02A .08B

.10C .06(60,000)

- Next, type the coefficient matrix.

1 -1 1 1 1 1 .02 .08 .10

- You can go three ways from here Cramer's rule,

setting up an inverse equation, or using rref. - Cramer's rule is very time consuming, however.

The inverse equation would look something like

this

- 1 -1 1
- 1 1 1
- .02 .08 .10

matrix A

matrix B

A B C

5000 60,000 3600

Multiply matrix A's inverse and matrix B (the

order is important) to get the values for A, B,

and C.

- The third method to solving systems with matrices

is by using rref, also known as augmented

matrices.

Surprise! It's a 3x4 matrix now. The fourth

column is the same as the answer column.

1 -1 1 5,000 1 1 1 60,000 .02 .08 .10 3,600

Go to your matrix window and hit the left arrow

to go to the MATH column. Go down until you see

B rref . Hit that and then choose your augmented

matrix. Hit enter and your answers should be

there.

You should've gotten this

- 1 0 0 21,875
- 0 1 0 32,500
- 0 0 1 5,625

x is 21,875 y is 32,500 z is 5,625

Solving Matrices using addition - Tip-

- Given
- A ,B
- AB

5 -3 4 3

- 1 3
- 2 -1

6 0 6 2

1 5 3 (-3) 2 4 (-1)

3

You can only add matrices together if A and B or

more is in the same order. Ex 2 by 2 pairs up

with another 2 by 2

Laws of Matrix -Tip-

- (1) A B B A
- (2) A (B C) (A B) C
- (3) 0A 0, where 0 is the zero matrix.
- (4) A 0 A.
- (5) A(B C) AB AC
- (6) A (BC) (AB)C
- (7) If A-1 and B-1 exist then (AB)-1 B-1

A-1 - Note These are other ways of writing matrices in

which the operations will make sense either way

Solving for X for Matrix Equations -Tip-

- Finding solutions for Ax B
- (A-1 A) X A-1B (1.) known as the

associative Law) - IX A-1B (2.) Definition of Inverse
- XA-1 B (3.) Definition of identity

note A-1 is never written as 1/A