Given the Characteristic Polynomial, Find the Rank of the Matrix

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)

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.

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