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MATRICES

WELCOME

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M TRIX A rectangular arrangement ofnumbers in rows and columns.

The OR ERof a matrix is the number of therows and columns.

The ENTRIESare the numbers in the matrix.

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67237

89511

36402

0759

3410

200318

20

11

6

0

7

9

3 x 3

3 x 5

2 x 2 4 x 1

1 x 4

(or squarematrix)

(Also called a rowmatrix)

(or squarematrix)

(Also called acolumn matrix)

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IfAand Bare both mn matrices then the sumofAand B, denotedA+ B, is amatrix obtained by adding correspondingelements ofAand B.

310

221A

412

403B

102

622BA

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

232

451

704

831

605

429

2833)2(1

)4(65015740249

603

1054

325

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In matrix algebra, a real number is often called a SCALAR. Tomultiply a matrix by a scalar, you multiply each entry in thematrix by that scalar.

)1(4)4(4

)0(4)2(4

14

024

416

08

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ABBA

CBACBA )()(

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IfAis an mn matrix and sis a scalar, then we let kA denote the matrix obtainedby multiplying every element ofA by k. This procedure is called scalar

multiplication.

k hA kh A

k h A kA hA

k A B kA kB

310

221A

930

663

331303

2323133A

PROPERTIES OF SCALAR MULTIPLICATION

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The mnzero matrix, denoted 0, is the mn

matrix whose elements are all zeros.

000)(

0

A

AA

AA

00

00 000

2 21 3

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BCACCBAACABCBA

CABBCA

PROPERTIES OF MATRIX

MULTIPLICATION

BAAB

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an nnmatrix with ones on the main diagonal and zeros

elsewhere

100

010

001

3

I

What isAI?

What is IA?

322

510

212

A

100

010

001

3I

A

322

510

212

A

322

510

212

Multiplying a matrix by

the identity gives the

matrix back again.

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IfAhas an inverse we say thatAis nonsingular.

IfA-1does not exist we sayAis singular.

To find the inverse of a matrix we put the matrix A, a line and then the

identity matrix. We then perform row operations on matrix A to turn it into

the identity. We carry the row operations across and the right hand side

will turn into the inverse.

To find the inverse of a matrix we put the matrix A, a line and then the

identity matrix. We then perform row operations on matrix A to turn it into

the identity. We carry the row operations across and the right hand side

will turn into the inverse.

72

31A

1210

0131

2r

1+r

2

1072

0131

12100131r2

1210

3701r1 r2

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bx 1 A

bx 11 AAA

bxA left multiply both sides by theinverse of A

This is just the identity

bx 1 AI

but the identity times a matrix

just gives us back the matrix

so we have:

This then gives us a formula for

finding the variable matrix: MultiplyA

inverse by the constants.

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Remember You cant multiply anymatrix. There are some conditionsThe number of columns on the first matrix has to be

same as the number of rows of the second matrix.

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The Transpose of a Matrix Sometimes it is of interest to interchange the rows

and columns of a matrix

The transpose of a matrix A=Aijis a matrix formedfrom A by inter changing rows and columns suchthat row i of A becomes columns I of the transposematrix. The transpose is denoted by At and

At=Aji when A= Aij

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Example of the Transpose A= 1 3 AT= 1 2

2 5 3 5

A= 1 3 4 AT= 1 00 1 0 3 1

4 0

It will be observed that ifAis m x n,At isn x m

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Q. What is a Part i t ioned Matr ixand what

does it have to do with me?

A. Ah, good question.

Well, a Part i t ioned Matr ixis a matrix that has

been broken down into several smaller matrices

But why tell you when I can show you a picture.

Lets say I have a 5x4 Matrix called G

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And now a partitioned version (with the partitionlines in red):

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And now we name the individual parts

(AKA: Blocksor Submatr ices):

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Now we can rewrite G as a 3x2

Matrix:

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1.You should press thecalculator thrice

You should the result as 100The only condition is that you

should not press 0 button

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2. A 40kg stone is broken into 4pieces.

What will be the wait of 4

broken pieces.You have to tell the wait of 4

pieces provided that the values

must satisfies every kg upto 40 kg

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V.VijayII M.sc. Mathematics120814

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