Linear Algebra - Diagonalizable and Eigenvalue Proof

[B_Phoenix]

Homework Statement

"Let A be a diagonalizable n by n matrix. Show that if the multiplicity of an eigenvalue lambda is n, then A = lambda i"

Homework Equations

The Attempt at a Solution

I had no idea where to start.

 

[VKint]

Since $$A$$ is diagonalizable, we can choose some invertible matrix $$S$$ such that $$A = S D S^{-1}$$, where $$D$$ is diagonal and the diagonal entries of $$D$$ are the eigenvalues of $$A$$. We can translate the assumption regarding the multiplicity of $$\lambda$$ into a statement about $$D$$, after which the result follows by using $$A = S D S^{-1}$$.