Diagonalize a $2 \times 2$ matrix and compute $A^n$

Primary Tool

Math Workspace (CAS)

CAS is the right tool because the page can move from eigenvalue computation to matrix reconstruction without splitting the work across separate tools.

Problem

Diagonalize $A = [4213]$ and use the result to express $A^{n}$ .

This is an important worked example because it turns the abstract definition of eigenvectors into a practical matrix factorization.

Why $A^{n}$ becomes manageable

Once you have $A = P D P^{- 1}$ , repeated multiplication gives $A^{n} = P D^{n} P^{- 1}$ . The hard work is all front-loaded into finding the eigenstructure once.

For a diagonal matrix, taking powers is immediate: every diagonal entry is raised to the $n$ th power independently.

Step-by-step walkthrough

The purpose of the computation is not just to find eigenvalues. It is to reorganize the matrix into coordinates where repeated multiplication is easy.

1Find the eigenvalues of $A = [4213]$ , which are $5$ and $2$ .
2For each eigenvalue, solve $(A - λ I) v = 0$ to get an eigenvector. One convenient choice is $v_{1} = [11]$ for $λ = 5$ and $v_{2} = [1 - 2]$ for $λ = 2$ .
3Place the eigenvectors into the matrix $P = [11 1 - 2]$ and place the eigenvalues on the diagonal of $D = [5002]$ .
4Check that $A = P D P^{- 1}$ by multiplying the three matrices back together.
5Use the factorization to write $A^{n} = P D^{n} P^{- 1}$ .
6Because $D$ is diagonal, compute $D^{n}$ by raising each diagonal entry independently: $D^{n} = [5^{n} 0 0 2^{n}]$ .

Common Pitfall

Diagonalization is not just renaming the matrix. It rewrites the problem so that powers of $A$ come from powers of the diagonal entries of $D$ .

Try a Variation

Try a matrix with repeated eigenvalues. When does diagonalization fail, and what part of the construction breaks?

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ExplanationLinear AlgebraMath Workspace (CAS)

What eigenvectors and eigenvalues mean geometrically

An eigenvector is a direction that survives a linear transformation without rotating away from its own line. The eigenvalue tells you the scale and possible sign flip along that direction.

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ProofLinear AlgebraProof Builder

Proof that the kernel of a linear map is a subspace

This is a standard linear algebra proof because it packages the subspace test into one reusable pattern: show zero is inside, then check closure under addition and scalar multiplication.

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