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.

A Matrix Representation of a Linear Transformation and Related Subspaces
Let $T:\R^4 \to \R^3$ be a linear transformation defined by
\[ T\left (\, \begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix} \,\right) = \begin{bmatrix}
x_1+2x_2+3x_3-x_4 \\
3x_1+5x_2+8x_3-2x_4 \\
x_1+x_2+2x_3
\end{bmatrix}.\]
(a) Find a matrix $A$ such that […]

Orthonormal Basis of Null Space and Row Space
Let $A=\begin{bmatrix}
1 & 0 & 1 \\
0 &1 &0
\end{bmatrix}$.
(a) Find an orthonormal basis of the null space of $A$.
(b) Find the rank of $A$.
(c) Find an orthonormal basis of the row space of $A$.
(The Ohio State University, Linear Algebra Exam […]

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 […]

Find Matrix Representation of Linear Transformation From $\R^2$ to $\R^2$
Let $T: \R^2 \to \R^2$ be a linear transformation such that
\[T\left(\, \begin{bmatrix}
1 \\
1
\end{bmatrix} \,\right)=\begin{bmatrix}
4 \\
1
\end{bmatrix}, T\left(\, \begin{bmatrix}
0 \\
1
\end{bmatrix} \,\right)=\begin{bmatrix}
3 \\
2 […]

Rank and Nullity of Linear Transformation From $\R^3$ to $\R^2$
Let $T:\R^3 \to \R^2$ be a linear transformation such that
\[ T(\mathbf{e}_1)=\begin{bmatrix}
1 \\
0
\end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix}
0 \\
1
\end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix}
1 \\
0
\end{bmatrix},\]
where $\mathbf{e}_1, […]

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