Can the following matrix be diagonalised?$$A=\left( \begin{array}{rrr}1&3&4\0&-1&3\0&0&-2\end{array}\right) $$

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BlackTom AI · Accepted Answer

First, note the given question asks whether the matrix can be diagonalised. The matrix A is upper triangular with diagonal entries 1, -1, and -2.

Key idea: a square matrix is diagonalizable over the reals (or complex numbers) if it has a full set of linearly independent eigenvectors, which is guaranteed when the matrix has distinct eigenvalues.

Option assessment context: the problem’s answer field provides a single option labeled 'a. Yes.' but the provided answer_options field is empty, so there are no alternative choices to evaluate in this setup. Despite that, we can still determine diagonalizability from the matrix itself.

Why diagonalizable here: the eigenvalues of A are precisely its diagonal entries for an upper triangular matrix, i.e., λ1 = 1, λ2 = -1, λ3 = -2. All three eigenvalues are distinct. A matrix with n distinct eigenvalues has n linearly independent eigenvectors, which means A is diagonalizable.

Constructive idea (optional deeper check): for a diagonalizable matrix, one can form P from a basis of eigenvectors and D as the diagonal matrix of eigenvalues, with A = P D P^{-1}. To illustrate, you would solve (A - λI)x = 0 for each λ ∈ {1, -1, -2} to obtain three independent eigenvectors, then assemble P from those eigenvectors. Since the eigenvalues are distinct, these eigenvectors are guaranteed to be independent and span the space.

Edge considerations: even if the matrix were not strictly triangular, the criterion of distinct eigenvalues would still guarantee diagonalizability. Here, the distinct eigenvalues provide a straightforward justification.

In summary, given A’s distinct eigenvalues, the matrix is diagonalizable over the real numbers. The provided answer choice 'a. Yes.' correctly reflects this conclusion, although the options list in the prompt is empty to compare against.

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

Can the following matrix be diagonalised?\[A=\left( \begin{array}{rrr}1&3&4\\0&-1&3\\0&0&-2\end{array}\right) \]

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