GATE CS 2018 Set 1 — Question 36
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Engineering Mathematics → Linear Algebra → Eigenvalues & Eigenvectors
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Question
Consider 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 diagonalizedWhich one of the following options is correct?
(I) P does not have an inverse
(II) P has a repeated eigenvalue
(III) P cannot be diagonalizedWhich one of the following options is correct?
Correct answer
(D) Only II and III are necessarily true
Solution
Given that the only eigenvectors of the matrix P are multiples of a single vector, the geometric multiplicity of its eigenvalue(s) is 1.
1.Statement (II): For a matrix, if there is only one linearly independent eigenvector, the algebraic multiplicity of the eigenvalue must be 2 (it is a repeated eigenvalue). Thus, (II) is necessarily true.
2.Statement (III): A matrix is diagonalizable if and only if it has a complete set of linearly independent eigenvectors (i.e., geometric multiplicity equals algebraic multiplicity for all eigenvalues). Here, the geometric multiplicity (1) is less than the algebraic multiplicity (2). Therefore, P cannot be diagonalized. Thus, (III) is necessarily true.
3.Statement (I): A matrix is invertible if its eigenvalues are non-zero. The repeated eigenvalue could be any value, including non-zero values (e.g., ). If , the matrix has an inverse. Thus, (I) is not necessarily true.
Therefore, only II and III are necessarily true.Continue learning with Success Tracker
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