Given the Characteristic Polynomial, Find the Rank of the Matrix

Ohio State University exam problems and solutions in mathematics

Problem 484

Let $A$ be a square matrix and its characteristic polynomial is give by
\[p(t)=(t-1)^3(t-2)^2(t-3)^4(t-4).\] Find the rank of $A$.

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

Note that the degree of the characteristic polynomial $p(t)$ is the size of the matrix $A$.
Since the degree of $p(t)$ is $3+2+4+1=10$, the size of the matrix $A$ is $10\times 10$.

From the characteristic polynomial, we see that the eigenvalues of $A$ are $1,2,3,4$.
In particular, $0$ is not an eigenvalue of $A$.
Hence the null space of $A$ is zero dimensional, that is, the nullity of $A$ is $0$.

By the rank-nullity theorem, we have
\[\text{rank of $A$} +\text{ nullity of $A$}=10.\] Hence the rank of $A$ is $10$.

Final Exam Problems and Solution. (Linear Algebra Math 2568 at the Ohio State University)

This problem is one of the final exam problems of Linear Algebra course at the Ohio State University (Math 2568).

The other problems can be found from the links below.

  1. Find All the Eigenvalues of 4 by 4 Matrix
  2. Find a Basis of the Eigenspace Corresponding to a Given Eigenvalue
  3. Diagonalize a 2 by 2 Matrix if Diagonalizable
  4. Find an Orthonormal Basis of the Range of a Linear Transformation
  5. The Product of Two Nonsingular Matrices is Nonsingular
  6. Determine Wether Given Subsets in ℝ4 R 4 are Subspaces or Not
  7. Find a Basis of the Vector Space of Polynomials of Degree 2 or Less Among Given Polynomials
  8. Find Values of $a , b , c$ such that the Given Matrix is Diagonalizable
  9. Idempotent Matrix and its Eigenvalues
  10. Diagonalize the 3 by 3 Matrix Whose Entries are All One
  11. Given the Characteristic Polynomial, Find the Rank of the Matrix (This page)
  12. Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$
  13. Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$

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