门| GATE-CS-2015(Set 3)|第65章

在给定矩阵中，特征值之一为1。对应于特征值1的特征向量为

⎡ 1 -1  2 ⎤
⎢ 0  1  0 ⎥
⎣ 1  2  1 ⎦

(A) A
(B) B
(C) C
(D) D答案： (B)
说明：让z代表特征值。

And let the given matrix be A (square matrix of order 3 x3)

The characteristic equation for this is :

AX = zX ( X is the required eigenvector )
AX - zX = 0
[ A - z I ] [X] = 0 ( I is an identity matrix of order 3 )

put z = 1 ( because one of the eigenvalue is 1 ) 

[ A - 1 I ] [X] = 0

The resultant matrix is :

[ 0 -1 2 ] [x1] [0]
| 0 0 0 ] |x2] =|0|
[ 1 2 0 ] |x3] [0]

Multiplying thr above matrices and getting the equations as:

-x2 + 2x3 = 0 ----------------(1)
x1 + 2x2 = 0-----------------(2) 

now let x1 = k, then x2 and x3 will be -k/2 and -k/4 
respectively.

hence eigenvector X = { (k , -k/2, -k/4) } where k != 0

put k = -4c ( c is also a constant, not equal to zero ), 
we get X = { ( -4c, 2c, 1c ) }, i.e. { c ( -4, 2, 1 ) }

Hence option B.

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