Let 𝐴 𝑏𝑒 𝑎𝑛 𝑛 × 𝑛 matrix. The characteristic polynomial of A is a polynomial of degree

1. Break Down the Problem

To determine the degree of the characteristic polynomial of an $n \times n$ matrix $A$, we need to understand the relationship between matrices and their characteristic polynomials.

2. Relevant Concepts

The characteristic polynomial of a matrix $A$ is defined as: $p(\lambda) = \text{det}(A - \lambda I)$ where $I$ is the identity matrix of the same dimension as $A$, and $\lambda$ is a scalar.

3. Analysis and Detail

1. The matrix $A$ is $n \times n$, meaning it has $n$ rows and $n$ columns.
2. $A - \lambda I$ will also be an $n \times n$ matrix.
3. The determinant of an $n \times n$ matrix is a polynomial in $\lambda$ of degree $n$. This is because the highest degree term in the determinant comes from the product of the diagonal elements when $\lambda$ is treated as a variable.

4. Verify and Summarize

Thus, we can conclude that the characteristic polynomial $p(\lambda)$ of the matrix $A$ is a polynomial of degree $n$. This confirms our understanding of the relationship between the structure of matrices and their characteristic polynomials.

Final Answer

The characteristic polynomial of an $n \times n$ matrix $A$ is a polynomial of degree $n$.