APLIMAT

Matrix

• Matrix is defined as a rectangular array of numbers

• A table of number or set of number

• The numbers inside the matrix is called elements or entries.

Matrix

2 1

3 5A2 X 2

Row

Column

Row

Column

X5 X 2

5 9

7 2

1 3

11 0

14 25

Matrix (Addition & Subtraction) 3 -1

2 0A

+

-7 2

3 5B

A B-4 1

5 5

A B- -10 -3

-1 -5

Matrix (Multiplication)

1 2

3 4 x5 6

7 8

19 22

43 50

Solution:

(5 + 14) (6 + 16)

(15 + 28) (18 + 32)

19 2243 50

(1 x 5) + (2 x 7) (1 x 6) + (2 x 8)

(3 x 5) + ( 4 x 7) (3 x 6) + ( 4 x 8)

Matrix (Multiplication)

3 1 2-2 0 5

-1 30 52 5

x1 24

12 19

Identity Matrix

1 00 1

1 0 00 1 00 0 1

1 0 0 00 1 0 00 0 1 00 0 0 1

Multiplying a matrix to its inverse will give the following matrices:

Inverse Matrix (Part 1)

• Formula:

A-1___________________

1ad - bc

A a bc d

d -b-c a

Inverse Matrix (Part 1)

A ad - bc

A-1___________________

1 d -b-c a

A

Inverse Matrix (Part 1)

B 3 -42 -5

B-1

___________________1

(3*-5)-(-4*2)

-5 4-2 3

Inverse Matrix (Part 1)

B-1 5/7 -4/7

2/7 -3/7

5/7 -4/72/7 -3/7 x 3 -4

2 -5

7 00 7 OR

1 00 1

Inverse Matrix (Part 2)

1 0 1

0 2 1

1 1 1AMatrix of Minor

2 1

1 1

0 1

1 1

0 2

1 1

0 1

1 1

1 1

1 1

1 0

1 1

0 1

2 1

1 1

0 1

1 0

0 2

Inverse Matrix (Part 2)

1 -1 -2

-1 0 1

-2 1 2

+ - +

- + -

+ - +

Apply the signs

Matrix of Cofactors

1 1 -2

1 0 -1

-2 -1 2

1 1 -2

1 0 -1

-2 -1 2

Adjecate(A) =

The diagonal stays the same.

Inverse Matrix (Part 2)

A 1*1 + 0*1 + 1*-2

-1AA -1

-1 -1 2

-1 0 1

-2 1 -2