Diagonalizable Matrix $A$ with Eigenvalues scalar=c, is $A=cI$

Question

Let $A$ be an $n \times n$ diagonalizable matrix such that all of it's eigenvalues are equal to a scalar, $c.$ Then $A=cI$.

Is this true? Why or why not? I'm thinking not, because I can come up with examples of matrices that are not diagonalizable that satisfies this. I am having a hard time coming up with a good proof for why it cannot be diagonalizable.

Answer 1

If $A$ diagonizable and all eigenvalues are equal, Jordan decomposition implies that there is an invertible matrix $M$ such that $$M^{-1}AM=\lambda E,$$ where $E$ denotes the identity matrix and $\lambda$ is the eigenvalue of $A$. This implies $$A=M(\lambda E)M^{-1}=\lambda MM^{-1}=\lambda E.$$

Answer 2

Let the only eigen value of $A$ be $\lambda.$ Since $A$ is diagonalizable it's minimal polynomial contains distinct linear factors having eigen values of $A$ as it's roots. In this case the minimal polynomial is $x - \lambda.$ Since minimal polynomial of $A$ annihilates $A$ it follows that $A=\lambda I,$ as claimed.