The Matrix Reloaded 00011000101001001110101001110001100010100100111010100111...

The Matrix Reloaded

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11101010101010101

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1

Matrix 2: release date May 2003

MathScience Innovation CenterB. Davis

The Matrix Reloaded B. Davis MathScience Innovation Center

Let’s review Inverses and Identities

If an expression is operated on by value x and the expression remains the same, then x is called a(n) _______________

for that operation.



If an expression is operated on by value x and the expression remains the same, then x is called a(n) _______________

for that operation.

Identity

For real numbers, the identity element for addition is___?

For real numbers, the identity element for multiplication is___?

0

1



In matrix addition, the identity matrix size must be:____________________The same as the addend

size

only zeros

In matrix addition, the identity matrix

must be filled with:_______________

Example: [ 2 4 ] + [ 0 0 ] = [ 2 4 ]


And Multiplication...In matrix multiplication, the identity matrix size must be:____________________

A square matrix.

a diagonal of 1’s and all the rest zeros

In matrix multiplication, the identity matrix

must be filled with:________________

78

32

78

32*

10

01

10

01*

78

32


And now…InversesIn matrix addition, inverse matrices are 2 matrices that add up to an identity matrix of all zeros.

In matrix addition, inverse matrices are composed elements that are the additive inverses of of elements in the original matrix.

Example: [ 2 4 ] + [ -2 -4 ] = [ 0 0 ]


And now… MultiplicationIn matrix multiplication, inverse matrices are 2 matrices whose product is an identity matrix of 0’s and 1’s.

By far, the easiest way to create an inverse matrix is A-1 on your TI-83plus.


Multiplicative InversesSteps:

1. Enter your matrix using MATRIX EDIT. 2. On the home screen, use MATRIX NAME (select yours) then press x-1.

Try this using [A] =

[A] -1 = ___?

61

42

8

1

16

14

1

8

3


Multiplicative Inverses

Now... this using [A] =

Try [A] [A] -1 = ___?

61

42

8

1

16

14

1

8

3

61

42* =

10

01


Multiplicative Inverses

Therefore, since their product is the identity matrix I2x2,

A and A-1 are called inverses.

61

42

8

1

16

14

1

8

3

61

42

8

1

16

14

1

8

3

= =

10

01


Time to learn how to do it without technology

Let’s start with [A]=Write it down.Now, here is the rule:

dc

ba

ac

bd

bcaddc

ba 11

Where ad -bc is called the determinant.

There are other rules

for larger matrices,

but 2 x 2 is all you

need to know!


without technology

Let’s start with the same [A].Write it down.

61

42

Now, the determinant is _____?

(Check on your calculator using

Matrix Math det [A]. )

-16


without technology

Next , let’s find the matrix

ac

bd

If [A] = then this new matrix

is formed by switching the 2 and the -6and then turning the 4 and the 1 negative.

61

42

61

42


without technology


ac

bd



61

42

61

42

21

46


without technology


ac

bd



61

42

21

46

21

46


without technology

We still are not finished!

ac

bdWe still need to multiply

by 1/det.Do you remember what the det was?

21

46

-16

So, multiply 1/-16 by and that is it!


without technology

Therefore:

1

61

42

16

2

16

116

4

16

6

21

46

16

1=

8

1

16

14

1

8

3


Your turn to try it !

Here is the rule:

ac

bd

bcaddc

ba 11

And here is your matrix [A]:

43

21


Your turn to try it !

What is your determinant?

2

1

2

312

2

1

2

32

2

2

4

13

24

64

1

43

211

What is your inverse matrix [A]-1?

43

21

-2

The Matrix Reloaded 00011000101001001110101001110001100010100100111010100111...

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