MATRIX:MATRIX: A rectangular A rectangular arrangement of arrangement of numbers in rows and numbers in rows and columns.columns.

The The ORDERORDER of a matrix of a matrix is the number of the is the number of the rows and columns.rows and columns.

The The ENTRIESENTRIES are the are the numbers in the matrix.numbers in the matrix.

502

126rows

columns

This order of this matrix This order of this matrix is a 2 x 3.is a 2 x 3.

A

a11 ,, a1n

a21 ,, a2n

am1 ,, amn

Aij

A matrix is any doubly subscripted array of elements arranged in rows and columns.

67237

89511

36402

3410

200

318 0759

20

11

6

0

7

9

3 x 3

3 x 5

2 x 2 4 x 1

1 x 4

(or square matrix)

(Also called a row matrix)

(or square matrix)

(Also called a column matrix)

Row VectorRow Vector

[1 x n] matrix

 

jn aaaaA ,, 2 1

Column VectorColumn Vector

i

m

a

a

a

a

A 2

1

[m x 1] matrix

Identity MatrixIdentity Matrix

I

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

Square matrix with ones on the diagonal and zeros elsewhere.

Transpose MatrixTranspose Matrix

A'

a11 a21 ,, am1

a12 a22 ,, am 2

a1n a2n ,, amn

Rows become columns and columns become rows

To add two matrices, they must have the same To add two matrices, they must have the same order. To add, you simply add corresponding order. To add, you simply add corresponding entries.entries.

34

03

12

70

43

35

)3(740

0433

13)2(5

44

40

23

9245

3108

2335

2571

)1(8 70 51 23

55 34 32 )2(9 =

= 7 7 4 5

0 7 5 7

To subtract two matrices, they must have the same To subtract two matrices, they must have the same order. You simply subtract corresponding entries.order. You simply subtract corresponding entries.

232

451

704

831

605

429

2833)2(1

)4(65015

740249

603

1054

325

724

113

810

051

708

342

=

5-2

-4-1 3-8

8-3 0-(-1) -7-1

1-(-4)

2-0

0-7

=

2 -5 -5

5 1 -8

5 3 -7

In matrix algebra, a real number is often called a In matrix algebra, a real number is often called a SCALARSCALAR. . To multiply a matrix by a scalar, you multiply each entry in To multiply a matrix by a scalar, you multiply each entry in the matrix by that scalar. the matrix by that scalar.

14

024

416

08

)1(4)4(4

)0(4)2(4

86

54

30

212

)8(360

52412

-2

6

-3 3

-2(-3)

-5

-2(6) -2(-5)

-2(3) 6 -6

-12 10

Matrix MultiplicationMatrix Multiplication

Matrices A and B have these dimensions:

[r x c] and [s x d]

Matrix MultiplicationMatrix Multiplication

Matrices A and B can be multiplied if:

[r x c] and [s x d]

c = s

Matrix MultiplicationMatrix Multiplication

The resulting matrix will have the dimensions:

[r x c] and [s x d]

r x d

232

451

704

831

605

429

232

451

704

7.06.04.0

AAthenIfA *..,

232

451

704

We can only find the We can only find the determinant of a square determinant of a square matrixmatrix

The value of the The value of the determinant for a 2×2 determinant for a 2×2 matrix is: det A = m11 matrix is: det A = m11 m22 - m12 m 21m22 - m12 m 21

rows

columns

Find the determinant Find the determinant value of the matrixvalue of the matrix

2221

1211

mm

mmLetA

20

11

831

605

429

LetZ

44

40

23

LetB

3410

200

318

LetR

Matrix Operations in ExcelMatrix Operations in Excel

Select the cells in which the answer will appear

Matrix Multiplication in ExcelMatrix Multiplication in Excel

1) Enter “=mmult(“

2) Select the cells of the first matrix

3) Enter comma “,”

4) Select the cells of the second matrix

5) Enter “)”

Matrix Multiplication in ExcelMatrix Multiplication in Excel

Enter these three key strokes at the same time:

control

shift

enter

232

451

704

831

605

429

Thank you