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

Linear Algebra exam problems and solutions at University of California, Berkeley

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.

(UCB-University of California, Berkeley, Exam)

LoadingAdd to solve later

Hint.

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.

Related Question.

A related question is:

What is the nullity of $A$?

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


LoadingAdd to solve later

More from my site

  • Inequality Regarding Ranks of MatricesInequality Regarding Ranks of Matrices Let $A$ be an $n \times n$ matrix over a field $K$. Prove that \[\rk(A^2)-\rk(A^3)\leq \rk(A)-\rk(A^2),\] where $\rk(B)$ denotes the rank of a matrix $B$. (University of California, Berkeley, Qualifying Exam) Hint. Regard the matrix as a linear transformation $A: […]
  • A Matrix Having One Positive Eigenvalue and One Negative EigenvalueA Matrix Having One Positive Eigenvalue and One Negative Eigenvalue Prove that the matrix \[A=\begin{bmatrix} 1 & 1.00001 & 1 \\ 1.00001 &1 &1.00001 \\ 1 & 1.00001 & 1 \end{bmatrix}\] has one positive eigenvalue and one negative eigenvalue. (University of California, Berkeley Qualifying Exam Problem)   Solution. Let us put […]
  • Square Root of an Upper Triangular Matrix. How Many Square Roots Exist?Square Root of an Upper Triangular Matrix. How Many Square Roots Exist? Find a square root of the matrix \[A=\begin{bmatrix} 1 & 3 & -3 \\ 0 &4 &5 \\ 0 & 0 & 9 \end{bmatrix}.\] How many square roots does this matrix have? (University of California, Berkeley Qualifying Exam)   Proof. We will find all matrices $B$ such that […]
  • Prove that the Length $\|A^n\mathbf{v}\|$ is As Small As We Like.Prove that the Length $\|A^n\mathbf{v}\|$ is As Small As We Like. Consider the matrix \[A=\begin{bmatrix} 3/2 & 2\\ -1& -3/2 \end{bmatrix} \in M_{2\times 2}(\R).\] (a) Find the eigenvalues and corresponding eigenvectors of $A$. (b) Show that for $\mathbf{v}=\begin{bmatrix} 1 \\ 0 \end{bmatrix}\in \R^2$, we can choose […]
  • Simple Commutative Relation on MatricesSimple Commutative Relation on Matrices Let $A$ and $B$ are $n \times n$ matrices with real entries. Assume that $A+B$ is invertible. Then show that \[A(A+B)^{-1}B=B(A+B)^{-1}A.\] (University of California, Berkeley Qualifying Exam) Proof. Let $P=A+B$. Then $B=P-A$. Using these, we express the given […]
  • If Column Vectors Form Orthonormal set, is Row Vectors Form Orthonormal Set?If Column Vectors Form Orthonormal set, is Row Vectors Form Orthonormal Set? Suppose that $A$ is a real $n\times n$ matrix. (a) Is it true that $A$ must commute with its transpose? (b) Suppose that the columns of $A$ (considered as vectors) form an orthonormal set. Is it true that the rows of $A$ must also form an orthonormal set? (University of […]
  • A Matrix Equation of a Symmetric Matrix and the Limit of its SolutionA Matrix Equation of a Symmetric Matrix and the Limit of its Solution Let $A$ be a real symmetric $n\times n$ matrix with $0$ as a simple eigenvalue (that is, the algebraic multiplicity of the eigenvalue $0$ is $1$), and let us fix a vector $\mathbf{v}\in \R^n$. (a) Prove that for sufficiently small positive real $\epsilon$, the equation […]
  • Find the Nullity of the Matrix $A+I$ if Eigenvalues are $1, 2, 3, 4, 5$Find the Nullity of the Matrix $A+I$ if Eigenvalues are $1, 2, 3, 4, 5$ 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 […]

You may also like...

1 Response

  1. 04/22/2017

    […] the post “Find the rank of the matrix $A+I$ if eigenvalues of $A$ are $1, 2, 3, 4, 5$” for a […]

Leave a Reply

Your email address will not be published. Required fields are marked *

More in Linear Algebra
Problems and Solutions of Eigenvalue, Eigenvector in Linear Algebra
Stochastic Matrix (Markov Matrix) and its Eigenvalues and Eigenvectors

(a) Let \[A=\begin{bmatrix} a_{11} & a_{12}\\ a_{21}& a_{22} \end{bmatrix}\] be a matrix such that $a_{11}+a_{12}=1$ and $a_{21}+a_{22}=1$. Namely, the sum...

Close