A repeated eigenvalue and a general power

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A-level extension. This is extension work, not a claim that Jordan form is required by your examination. Course filters use selected objective mappings; none of these tasks establishes complete paper coverage.

Before you start

Eigenvalues and eigenvectors. Read the relevant method, then decide which parts you can solve without hints.

Check a misconception first · Try a connected problem

Read an original example question

A = $\begin{pmatrix}1& 4\\0& 1\end{pmatrix}$ .

Part a

Determine whether A can be diagonalised. Justify your answer.

Part b

Find $A^{n}$ for positive integers n, choosing a suitable method.

Part c

Hence find the upper-right entry of $A^{3}$ .

Model solution and review criteria

Part a

The only eigenvalue is 1, repeated twice. (A − 1I)(x,y) = (4y,0), forcing y = 0. The eigenspace has dimension 1, so no invertible eigenvector basis exists.

Find the repeated eigenvalue.
Solve the eigenspace condition.
Explain why one direction is insufficient.

Part b

Write A = 1I + N, with N = $\begin{pmatrix}0&4\\0&0\end{pmatrix}$ and $N^{2}$ = 0. The binomial expansion gives $A^{n}$ = $\begin{pmatrix}1ⁿ& n4(1)^(n−1)\\0&1ⁿ\end{pmatrix}$ . It is not obtained from an invertible eigenbasis.

Introduce and verify the nilpotent part.
Apply the terminating binomial expansion.
State a formula valid for every positive n.

Part c

Use n = 3 in the general formula.

Use the general result consistently.

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