Download Determinants and matrices

Determinants and matrices
Definition :
A matrix is a rectangular array of numbers Hor functionsL
arranged in rows and columns
Example :
A=J
2 1 3
N
1 0 2
is a 2 ´ 3 H2 by 3L matrix because it has 2 rows and 3 columns. A m ´ n matrix is said to be of order m ´ n.
H1 2 3 L, order 1 ´ 3
jij 6 zyz, order 2 ´ 1
Column Matrix :
k1{
Square Matrix :
A matrix with same number of rows as columns *
Row Matrix :
Examples :
J
ex
a b
j
N, i
j 2x
c d
ke
3 xy
z
z
x2 {
are 2 ´ 2 square matrices OR square matrices of order 2.
The numbers Hor functionsL are called "entries" or elements " of the matrix.
Double suffix notation :
Consider m ´ n matrix A
a11
i
j
j
j
a21
j
j
A=j
j
j
j
j .
k am1
a12
a22
.
am2
. a1 n y
z
z
. a2 n z
z
z
z
z
. . z
z
z
. amn {
The element in ith row and j th column is denoted by
aij
where i = row number
j = column number
Determinants :
Example :
det J
Eg : Ë
a11
a21
With each square matrix we associate a number denoted by
det HaL or È aij È or È A È called the determinant of A
Determinant of order 2
a12
a
N = Ë 11
a22
a21
a12
Ë = a11 a22 - a21 a12
a22
3 -2
Ë = 3 ´ 5 - 4 ´ H-2L = 15 + 8 = 23
4 5
H a number L
Example :
Construct a 2 nd order det in which
aij = H1 + 3 iL - j2 and evaluate.
Clearly i = 1, 2,
a11
a12
a21
a22
j = 1, 2, thus
= H1 + 3L - 12 = 4 - 1 = 3
= H1 + 3L - 22 = 4 - 4 = 0
= H1 + 3 ´ 2L - 12 = 7 - 1 = 6
= H1 + 3 ´ 2L - 22 = 7 - 4 = 3
det Haij L = Ë
3 0
Ë = 3´3 - 6´0 = 9
6 3
Minors and Cofactors :
The minor Mij associated with element aij in an n th order det
is defined to be a det of order n - 1 obtained by removing
ith row and jth column of the original det.
Example :
a11
i
j
j
j
j
a21
j
j
j
k a31
The minor of a22 is Ë
a12 a13 y
z
z
z
a22 a23 z
z
z
z
a32 a33 {
a13
Ë
a33
a11
a31
Laplace Exapansion of Dets :
We know D = Ë
a11
a21
a12
Ë = a11 a22 - a21 a12
a22
Here c11 = a22 , c12 = a12 ,
c21 = -a12 ,
c22 = a11
Clearly D = a11 c11 + a12 c12 HExpansion by first rowL
Similarly
D = a11 c11 + a21 c21
D = a21 c21 + a22 c22
D = a12 c12 + a22 c22
H Expansion by first column L
H Expansion by 2 nd row L
H Expansion by 2 nd columnL