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Gate2018 cs Q49

49Consider a matrix P whose only eigenvectors are the multiples of
.
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?
  • Option : D
  • Explanation : Only Eigen vector is 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 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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