vj presentation.ppt
TRANSCRIPT
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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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MIND READER
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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