Induction Matrix
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Transcript of Induction Matrix
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Algebra
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Topics:
Matrix
Equality of Matrices
Types of Matrix
Matrix Operations
Determinants of Matrix
Carmers rules (Determinant Method)
Applications
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Matrix
A !
a11 ,~ , a1n
a21 ,~ , a2 n
~ ~ ~ ~am1 ,~ , amn
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! A ij_ a
A rectangular arrangement of mn numbersinto m horizontal rows and n vertical
columns enclosed by a pair of brackets [],
such as
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Equality of Matrices
Two M tr ces d B re s d to be equ lwr tte =B f they re of the s e order d
f ll correspo d g e tr es re equ l
-
!
432
015A
-
2232
0132
xB
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Row Matrix
[1 x n] matrx
? A _ ajn aaaaA ,,21 !~
A Matrx thathasexactlyonerow iscalled RowMatrix
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Column Matrix
_ ai
m
a
a
aa
A 2
1
!
-
~!
[m x 1] matrix
A Matrix consistingofa singlecolumn iscalledColumn Matrix
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Square Matrix
B ! 5 4 7
3 6 1
2 1 3
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Anm
Xnmatrix
is
said
to
be
square
matrix
ofordernifm=n
Samenumberofrowsandcolumns
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Identity Matrix
I !
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
-
ASquare
matrix
is
said
to
be
identity
matrix
orunitmatrixifallitsmaindiagonal
elementsare 1sandallotherentriesare 0s.
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Triangular Matrix
A square matrix issaidtobe an upper(lower)triangularmatrix ifalltheelements
below(above)the main diagonal arezeros.
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!
300
160
745
B
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!
342061
005
C
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Diagonal Matrix
A square matrix issaidtobediagonal ifeachofitsentries notfallingon the main
diagonal iszero.
-
!
300
000005
B
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Scalar Matrix
A diagonal matrix whoes allthediagonalelements areequal iscalled a Scalarmatrix
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!
500
050005
B
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Transpose Matrix
A' !
a11 a21 ,~, am1
a12 a22 ,~, am 2
~~~~~a1n a2n ,~, amn
-
Rows become columns and
columns become rows
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Matrix Operations Addition
Subtraction
Multiplication Inverse
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Addition
1
4
2
3
5
8
6
7+ =
A B+ =
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Addition ConformabilityTo add two matrices A and B:
# of rows in A = # of rows in B
# of columns in A = # of columns in
B
TwoMatricesofsameorder
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Subtraction
1
4
2
3
5
8
6
7 - =
B A- =
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Subtraction
1
4
2
3
5
8
6
7 - =
4
4
4
4
B A- = C
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Subtraction Conformability
To subtract two matrices A and B:
# of rows in A = # of rows in B
# of columns in A = # of columns in
B
TwoMatricesofsameorder
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Problem: 1
An automobile dealer sells two car models,
standard and deluxe. Each is available in one oftwo colors, white and red. His sales for the monthsof January & February are given by the matrices
Jan:
Feb:
Find his total sales for each model and color for bothmonths.
Standard Deluxe
White 2 1
Red 3 4
Standard Deluxe
White 3 1
Red 2 3
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Problem 2:
Three persons buy cold drinks of different
brands A, B & C. The first person buys 12bottles of A, 5 bottles of B & 3 bottles of C.The second person buys 4 bottles of A, 6bottles of B & 10 bottles of C. The third
person buys 6 bottles of A, 7 bottles of B &9 bottles of C. Represent these informationin the form of a matrix. If each bottle ofbrand A costs Rs.4, Each of B costs Rs. 5and each of C costs Rs. 6, then using matrix
operations. Find the total sum of moneyspent individually by the three persons forthe purchase of cold drinks.
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Multiplication Conformability Regular Multiplication
To multiply two matrices A and B:
# of columns in A = # of rows in B
Multiply: A (m x n) by B (n by p)
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Multiplication General Formula
Cij= 'A
ikx B
kjk=1
n
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MultiplicationI
1
4
2
3
5
8
6
7 x =
A Bx =
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MultiplicationII
1
4
2
3
5
8
6
7x =
A Bx = C
(5x1)
C11
='A11
xB11k=1
n
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MultiplicationIII
1
4
2
3
5
8
6
7x =
A Bx = C
(5x1)+(6x3)
C11
='A12
xB21k=2
n
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MultiplicationIV
1
4
2
3
5
8
6
7x =
A Bx = C
23 (5x2)+(6x4)
C12
='A1k
xBk2k=1
n
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MultiplicationV
1
4
2
3
5
8
6
7x =
A Bx = C
23
(7x1)+(8x3)
34
C21
='A2k
xBk1k=1
n
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MultiplicationVI
1
4
2
3
5
8
6
7x =
A Bx = C
23 34
(7x2)+(8x4)31
C22
='A2k
xBk2k=1
n
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MultiplicationVII
1
4
2
3
5
8
6
7x =
A Bx =
C
23 34
31 46
m x n n x p m x p
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DETERMINANTS OF A 3 X 3 MATRIX
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1 1 1
2 2 2
3 3 3
a b c
a b c
a b c
= +2 2
1
3 3
b ca
b c
1 1
2
3 3
b ca
b c
1 1
3
2 2
b ca
b c
= 1 2 3 3 2
a b c b c + 2 1 3 3 1
a b c b c 3 1 2 2 1
a b c b c
= 1 2 3 3 2a b c b c + + 2 1 3 3 1( 1)a b c b c 3 1 2 2 1a b c b c
=1 1
a A + +2 2
a A3 3
a A
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2 35
1 4
5 1 2
3 2 3
8 1 4
=
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2 35
1 4
5 1 2
3 2 3
8 1 4
=1 2
( 3)1 4
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2 35
1 4
5 1 2
3 2 3
8 1 4
=1 2
( 3)1 4
1 28
2 3
+
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2 35
1 4
5 1 2
3 2 3
8 1 4
=1 2
( 3)1 4
1 28
2 3
+
=
5 8 ( 3)
( 3) 4 2 +
8 3 ( 4)
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2 35
1 4
5 1 2
3 2 3
8 1 4
=1 2
( 3)1 4
1 28
2 3
+
=
5 8 ( 3)
( 3) 4 2 +
8 3 ( 4)
= 55 ( 6) + 56
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2 35
1 4
5 1 2
3 2 3
8 1 4
=1 2
( 3)1 4
1 28
2 3
+
=
5 8 ( 3)
( 3) 4 2 +
8 3 ( 4)
= 55 ( 6) + 56
= 117
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Problem 2:
The annual sale volumes of three products X, Y, Zwhoes sales prices per unit are Rs.3.50, Rs.2.75,Rs.1.50 respectively, in two differentmarkets I &II are shown below:
Product
Market X Y Z
I 6000 9000 13000
II 12000 6000 17000
Find the total revenue in each market with the helpof matrices.
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Unary Operations: Inverse
Inverse of (2 x 2) matrix
Find determinant
Swap a11 and a22
Change signs of a12 and a21 Divide each element by determinant
Check by pre- or post-multiplying by
inverse
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Inverse of 2 x 2 matrix
Find the determinant
= (a11 x a22) - (a21 x a12)
For
det(A) = (2x3) (1x5) = 1
2
3
5
1=A
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Inverse of 2 x 2 matrix
Swap elements a11 and a22Thus
becomes
2
3
5
1=A
3
2
5
1
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Inverse of 2 x 2 matrix
Change sign of a12 and a21Thus
becomes
3
2
5
1=A
3
2
-5
-1
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Inverse of 2 x 2 matrix
Divide every element by the determinant
Thus
becomes
(luckily the determinant was 1)
3
2
-5
-1=A
3
2
-5
-1
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Inverse of 2 x 2 matrix
Check results with A-1A = I
Thus
equals
3
2
-5
-1x
1
1
0
0
2
3
5
1
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CRAMERSRULE:
Its an application of determinants is to solve a
system of linear equations in which number ofvariables are equal to the number of equationsand the coefficient matrix of the system of
equations is non-singular
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Problem 3:
a) X - 3Y + 4Z = 3
2X 5Y + 7Z = 6
3X 8Y + 11Z = 11