Matrix Arithmetic

A matrix M is an array of cell entries (mrow,column) and it must have rectangular dimensions (Rows x Columns).

Example:

2

5 17 2 20

5 0 6 15

21 10

r

M x

r t g

3x4 2,4 :m

3

4

15xDimensions:

Aarow,column

A

2,4 :a

Matrix

Scalar Multiplication

3 2 54

8 3 1

12 8 20

32 12 4

Every entry in the matrix is multiplied by the number outside the matrix (scalar).

Example:

Matrix Addition/SubtractionIF the matrices have the same dimensions, add

or subtract corresponding cell entries.

Examples:

a b c g h i

d e f j k l

a g b h c i

d j e k f l

5 3

12 0

4 10

5 3

12 0

4 10

b+h

8

12

14

Matrix Addition/Subtraction

Perform the indicated operation:

3 0.4 0

8 7 4 18 2

z w

The matrices MUST have the

same dimensions!

Matrix Multiplication

4 52 1 3

1 34 2 1

2 1

A

2x33x2

1

1

2

2

2 4 1 1 3 2 2 5 1 3 3 1

4 4 2 1 1 2 4 5 2 3 1 1

15 16

20 272x2

1Multiply the elements of each row of the first matrix by the elements of each column in the second matrix. 2Add the products. 3The answer goes into arow of 1st, column of 2nd.

a1,1 a1,2

a2,1 a2,2

Matrix Multiplication

Can we multiply these…

8

8 .1 2 5 2

0 1 52 2 0

8 17 5 5 9

4

4 5

7 2 .75 1 3

2 1

2 1 3 8 7

4 2 1 5 2

?

2x3 2x2 3x45x1

1x33x2

# of columns in 1st MUST be the

same as # of rows in 2nd!

No No

Yes

Matrix Multiplication

2 3 1 2 11 16

4 5 3 4 19 28

6 4 7 5 4 3 74 78 72

4 8 5 4 3 3 72 70 66

5 6 6 4 6 6 73 74 69

5 34 13 0.30 5.10

4 3 3 0.45 4.35

4 6 6 0.60 7.50

6 4 7

1 1 1 4 8 5 15 18 18

5 6 6

179

(b)

180

(a)

182

183

(b)

3x3 3x3 3x3

1x3 3x3 1x3

2x2 2x2 2x2

3x3 3x1 3x1

2 2

3 3

1 1

3 3

3 3

3 3

2 2

1 1

The dimensions of a product of matrices are the # of rows of the first matrix by the # of columns of

the second matrix.

In order to multiply matrices, the # of columns in 1st matrix MUST be the same as # of

rows in 2nd Matrix.

Order in Matrix Multiplication Matters

6 4 7 5 4 3 74 78 72

4 8 5 4 3 3 72 70 66

5 6 6 4 6 6 73 74 69

5 4 3 6 4 7 61 70 73

4 3 3 4 8 5 51 58 61

4 6 6 5 6 6 78 100 94

E B

B E

Identity Matrix

The product of a square matrix A and its identity matrix I, on the left or the right, is A.

AI = IA =A

General Form:

1 0 0 0 0

0 1 0 0 0

0 0 0 1 0

0 0 0 0 1

I Must be a

square matrix

Identity Matrix Example

5 8 1

5 2 8

0 7 15

5 8 1

5 2 8

0 7 15

1 0 0

0 1 0

0 0 1