GATE Computer Science (CS) 2018 Shift 1 Solved Paper

Question : 65 of 65

Marks: +1, -0

Consider a matrix

{P}

whose only eigenvectors are the multiples of

[\table 1 ; 4]

.
Consider the following statements :
I. P does not have an inverse.
II. P has a repeated eigenvalue.
III. P cannot be diagonalized.
Which one of the following options is correct?

Only I and II are necessarily true
Only II is necessarily true
Only I and II are necessarily true
Only II and III are necessarily true

Validate

Solution:

Only Eigen vector is

[\table 1 ; 4]

multiples means that eigen value is repeated since if eigen values were distinct we will get one more independent eigen vector. So, II P has repeated eigen value is true. I need not be true since

[\table 2 , 0 ; 0 , 2]

has repeated eigen values and yet it is invertible. III is true since if matrix has repeated eigen values then it cannot be diagonalizable.

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