Consider a matrix P whose only eigenvectors are multiples of [1, 4]ᵀ. Consider the following stateme

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Question

Consider a matrix P whose only eigenvectors are multiples of [1, 4]ᵀ. 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?

Accepted Answer

Correct Answer: D — Only II and III are necessarily trueStep-by-step Solution:The matrix P is 2×2 and has only one independent eigenvector direction: all eigenvectors are scalar multiples of [1, 4]ᵀ.Statement II — P has a repeated eigenvalue (TRUE): If a 2×2 matrix had two distinct eigenvalues, it would necessarily have two linearly independent eigenvectors (one for each eigenvalue from distinct eigenspaces). Since P has only one independent eigenvector direction, it cannot have two distinct eigenvalues. Therefore P must have a repeated (degenerate) eigenvalue. Statement II is necessarily true.Statement III — P cannot be diagonalized (TRUE): A square matrix of order n is diagonalizable if and only if it possesses n linearly independent eigenvectors. Since P is 2×2 but has only 1 linearly independent eigenvector, it is defective and cannot be diagonalized. Statement III is necessarily true.Statement I — P does not have an inverse (NOT necessarily true): Invertibility depends on whether the eigenvalue is zero. A matrix with a repeated non-zero eigenvalue is still invertible. For example, P = [[1, 1], [0, 1]] (a Jordan block) has eigenvalue 1 repeated twice, with only one independent eigenvector [1, 0]ᵀ, yet det(P) = 1 ≠ 0 so P is invertible. The repeated eigenvalue could be any non-zero value, making invertibility possible. Statement I is not necessarily true.Key Concept: A matrix that has fewer linearly independent eigenvectors than its dimension is called defective. A defective 2×2 matrix necessarily has a repeated eigenvalue (Statement II is true) and cannot be diagonalized (Statement III is true). However, a repeated eigenvalue may be non-zero, so the matrix may still be invertible (Statement I is not necessarily true).

Consider a matrix P whose only eigenvectors are multiples of [1, 4]ᵀ. 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?

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Consider a matrix P whose only eigenvectors are multiples of [1, 4]ᵀ. 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?

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