Page 166 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Hence, latent vector corresponding to the latent root, = 15 will be
3
x 1 2k 3
x = x 2 2k 3
x k
3 3

Example 4: If a + b + c = 0, find the characteristic values of the matrix
a c b
c b a
b a c
Solution: We have the characteristic equation of A
|A – I| = 0

a c b a b c c b
c b a a b c b a
or
b a c a b c a c
On replacing C by C + C + C .
1 1 2 3
c b
= b a [ a + b + c = 0]
a c

c b
= 0 b c c b
0 a c c b

On operating R – R and R – R
2 1 3 1
2
2
2
2
= [(a + b + c – ab – bc – ca) – ]
But a + b + c = 0, i.e., (a + b + c) = 0
2
or a + b + c + 2ab + 2bc + 2ca = 0
2
2
2
1 2 2 2
or (ab bc ca ) (a b c )
2
 Characteristic equation becomes
1 2 2 2 2
2
2
2
(a b c (a b c ) 0
2
3 2 2 2 2
or (a b c ) 0
2
1/2
3 2 2 2
which gives = 0 or (a b c )
2
Example 5: If A be a square matrix, show that the characteristic values of the matrix A are
the same as those of its transpose A’.

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