Which of the following statements are true? Select all that apply.

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BlackTom AI · Accepted Answer

Question restatement: The prompt asks which of the following statements are true, and it is a multiple-answer question with two options.

Option a: "A matrix can only be diagonalised if the algebraic multiplicity is equal to the geometric multiplicity for every eigenvalue." This is the standard criterion for diagonalizability. For each eigenvalue, the dimension of its eigenspace (geometric multiplicity) must match its algebraic multiplicity (the multiplicity of the eigenvalue as a root of the characteristic polynomial). If even one eigenvalue has geometric multiplicity less than its algebraic multiplicity, you cannot form a full set of n linearly independent eigenvectors, so the matrix is not diagonalizable. Conversely, if this equality holds for all eigenvalues, you obtain a complete eigenbasis and can diagonalize the matrix. So this statement is a correct characterization and is true.

Option b: "A matrix with all distinct eigenvalues is always diagonalisable." When all eigenvalues are distinct (in the field you're working over), you get n linearly independent eigenvectors, one for each eigenvalue, which provides a basis of eigenvectors and yields a diagonalization. This is a well-known sufficient condition for diagonalizability. However, subtlety arises regarding the field: if the matrix is considered over a field where some eigenvalues are not contained in that field (for example, real matrices with complex eigenvalues), saying the matrix is diagonalisable may depend on extending the field (diagonalization may occur over the complex numbers but not over the reals). In the standard linear algebra context, if the eigenvalues are distinct within the field of interest, the matrix is diagonalisable in that field. Thus, the statement is true in the typical setting, though one should be mindful of the field in question.

Summary of reasoning: Option a presents a necessary-and-sufficient condition for diagonalizability, and Option b gives a widely cited sufficient condition (distinct eigenvalues imply diagonalizability) with a caveat about the underlying field. Both statements are correct within the standard interpretations, with the caveat for option b about where the eigenvalues lie and over which field diagonalization is considered.

MTH1030 -1035 - S1 2025 MTH1030/35 Week 6 lesson quiz: Eigenvectors and eigenvalues

Which of the following statements are true? Select all that apply.

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