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

Problem 35

Let $A$ be an $n$ by $n$ matrix with entries in complex numbers $\C$. Its only eigenvalues are $1,2,3,4,5$, possibly with multiplicities. What is the rank of the matrix $A+I_n$, where $I_n$ is the identity $n$ by $n$ matrix.

The problem is asking if you understand the definition of eigenvalues and basic relation between the rank and non-singularlity.

Solution.

The matrix $A+I_n$ is nonsingular, otherwise $-1$ is an eigenvalue of $A$ but by assumption it is impossible.
Since the matrix $A+I_n$ is nonsingular, it has full rank. Since $A+I_n$ is $n$ by $n$ matrix, its rank must be $n$.

Comment.

The solution is very short and simple. The point is to notice that $A+I_n$ is of the familiar form $A-\lambda I_n$.

If the problem asked to find the rank of $A-6I_n$, then it would have been easier to notice this.

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