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MATRICESCompiled by Dr. S.S. Chauhan
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An m x n matrix is an array of mxn objectsin m rows and n columns and is representedin the form
Definition
][
321
3333231
2232221
1131211
nmmnmmm
n
n
n
ij
aaaa
aaaa
aaaa
aaaa
a A
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3
Squarematrix:
If m = n , then the matrix is a square
matrixElements a ij for i=j called diagonalelements and
is called the trace of A.
Types of Matrices
...11 221
n
ii nn
i
a a a a
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Partitioned matrices:
2221
1211
34333231
24232221
14131211
A A
A A
aaaa
aaaaaaaa
A
submatrix
3
2
1
34333231
24232221
14131211
r
r r
aaaa
aaaaaaaa
A
432134333231
24232221
14131211
cccc
aaaa
aaaaaaaa
A
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Then (1) A + B = B + A
(2) A + ( B + C ) = ( A + B ) + C
(3) ( cd ) A = c ( dA )
(4) 1 A = A
(5) c ( A + B ) = c A + c B
(6) ( c + d ) A = cA + dA
scalar :, ,,, If d c M C B A nm
Properties of matrix addition and scalar multiplication:
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calar c M A nm s: , If
A A nm 0(1)Then
nm A A 0)((2)
nmnm or AccA 000)3(
Notes:
(1) 0 m n : the additive identity for the set of all m n matrices
(2) A : the additive inverse of A
Properties of zero matrices:
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A A T T )( )1(T T T B A B A )( )2(
)()( )3( T T AccA
)( )4( T T T A B AB
Properties of transposes:
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1nEliminatioJordan-Gauss || A I I A Ex 2: (Find the inverse of the matrix)
31
41 A
Sol:
I AX
10
01
31
41
2221
1211
x x
x x
10
01
33
44
22122111
22122111
x x x x
x x x x
Find the inverse of a matrix by Gauss-Jordan Elimination:
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110
301
031
141)1()4(
21)1(
12 , r r
110401
131041)2(
)4(21
)1(12 , r r
1,3 2111 x x
1,4 2212 x x
)( 11
43 1-2221
12111 AA I AX x x
x x A X
Thus
(2) 1304
(1) 0314
2212
2212
2111
2111
x x x x
x x x x
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Rank of a Matrix
A number r is said to be rank of amatrix A, if there existsa non zero minor of order r and all
minors oforder r+1 vanish.
Or equivalentlyThe maximum number of linearlyindependent rows.
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I A 0
(1)
0)( )2(factors
k A AA Ak
k
integers:, )3( sr A A A sr sr
rs sr A A )(
k n
k
k
k
n d
d
d
D
d
d
d
D
00
00
00
00
00
00
)4(2
1
2
1
Power of a square matrix:
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Thm 1: (Systems of equations with unique solutions)
If A is an invertible matrix, then the system of linear equations Ax = b has a unique solution given by
b A x 1
Pf:
( A is nonsingular)
b A x
b A Ix
b A Ax Ab Ax
1
1
11
This solution is unique.
.equationof solutions twowereandIf 21 b Ax x x
21then Axb Ax 21 x x (Left cancellation property)
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2.4 Elementary MatricesRow elementary matrix:
An n n matrix is called an elementary matrix if it can be obtained
from the identity matrix I n by a single elementary operation.
Three row elementary matrices:
)()1( I r R ijij )0( )()2( )()( k I r R k i
k i
)()3( )()( I r R k ijk
ij
Interchange two rows.Multiply a row by a nonzeroconstant.Add a multiple of a row toanother row.
Note:Only do a single elementary row operation.
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Notes:
A R Ar ijij )( )1(
A R Ar k ik
i)()( )( )2(
A R Ar k ijk
ij)()( )( )3(
EA Ar
E I r
)(
)(
Thm 2.12: (Representing elementary row operations)Let E be the elementary matrix obtained by performing anelementary row operation on I m . If that same elementary rowoperation is performed on an m n matrix A, then the resultingmatrix is given by the product EA .
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If A is an n n matrix, then the following statements are equivalent.
(1) A is invertible.
(2) A x = b has a unique solution for every n 1 column matrix b.
(3) A x = 0 has only the trivial solution.
(4) A is row-equivalent to I n .
(5) A can be written as the product of elementary matrices.
Thm 2.15: (Equivalent conditions)
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LU A L is a lower triangular matrix
U is an upper triangular matrix
If the n n matrix A can be written as the product of a lowertriangular matrix L and an upper triangular matrix U , thenA=LU is an LU-factorization of A
Note:
If a square matrix A can be row reduced to an upper triangularmatrix U using only the row operation of adding a multiple ofone row to another row below it, then it is easy to find an LU -factorization of A .
LU A
U E E E A
U A E E E
k
k
112
11
12
LU -factorization:
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Eigen Values and Eigen Vectors
Definition. A nonzero vector x is an eigenvector of asquare matrix A if there exists a scalar such that Ax = x . Then is an eigenvalue of A.
Note : The zero vector can not be an eigenvector eventhough A0 = 0 . But = 0 can be an eigenvalue.
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Properties of eigenvalues and eigenvectors Property 1: The sum of the eigenvalues ofa matrix equals the trace of the matrix.
Property 2: A matrix is singular if and onlyif it has a zero eigenvalue.
Property 3: The eigenvalues of an upper(or lower) triangular matrix are theelements on the main diagonal.
Property 4: If is an eigenvalue of A andA is invertible, then 1/ is an eigenvalueof matrix A -1 .
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Property 5: If is an eigenvalue of A then k isan eigenvalue of kA where k is any arbitrary
scalar.Property 6: If is an eigenvalue of A then k isan eigenvalue of Ak for any positive integer k .
Property 7: If is an eigenvalue of A then isan eigenvalue of AT.
Property 8: The product of the eigenvalues(counting multiplicity) of a matrix equals the
determinant of the matrix.
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