MATHEMATICS-I
7
–6+4=0 3
( – 4 ) – 2 ( - 2 ) = 0 2
( – 2 ) [ ( + 2 ) – 2 ] = 0 ( – 2 ) ( + 2 – 2 ) = 0 2
= 2,
-2±4+8 2 - 2 ± 23 2
= 2, - 1 ± 3
= 2,
The characteristic roots are
2,
- 1 ± 3
Example : 2
Find the eigen values of
Let A
a 0 0
=
a 0 0
h b 0
h b 0
g 0 c
g 0 c
The characteristic equation is | A - I | = 0 a- h 0 b- 0 0 i.e. (a - ) (b - ) (c - ) = 0 = a, b, c the eigen values are a, b, c (i.e )
g 0 c-
=
Example : 3 Prove that the matrices A, B, C given below have the same characteristic values.
A=
0 a b
a 0 c
b c 0
0 b a
B=
b 0 c
a c 0
C=
0 c b
=
0
c 0 c
c a 0
Solution : The characteristic equation of A is A - I = 0 |A-I|=0
- a B
a - c
b c -
- ( – c ) – a ( - a - bc ) + b (ac + b ) = 0 3 2 2 2 - + c + a + abc + abc + b = 0 3 2 2 2 - ( a + b + c ) – 2 abc = 0 2
2
The characteristic equation of B is | B - I | = 0 |B-I|=0 ;
0
- -
b c
c = 0
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