# Find the Nullity of the Matrix $A+I$ if Eigenvalues are $1, 2, 3, 4, 5$

## Problem 387

Let $A$ be an $n\times n$ matrix. Its only eigenvalues are $1, 2, 3, 4, 5$, possibly with multiplicities.

What is the nullity of the matrix $A+I_n$, where $I_n$ is the $n\times n$ identity matrix?

(The Ohio State University, Linear Algebra Final Exam Problem)

## Solution.

Recall that $\lambda$ is an eigenvalue of the matrix $A$ if and only if the matrix $A-\lambda I$ is singular.
Thus, $A+I$ is nonsingular, otherwise $-1$ is an eigenvalue of $A$.

Since $A+I$ is nonsingular, the we have the null space $\calN(A+I)=\{\mathbf{0}\}$, and hence the nullity of $A+I$ is zero.

## Related Question.

What is the rank of $A$?

See the post “Find the rank of the matrix $A+I$ if eigenvalues of $A$ are $1, 2, 3, 4, 5$” for a solution.

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1. 04/22/2017

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