How to find the characteristic polynomial of a matrix?
Understand the Problem
The question is asking for the method to derive the characteristic polynomial of a given matrix. This typically involves using the formula det(A - λI), where A is the matrix in question, λ is a scalar (the eigenvalue), and I is the identity matrix.
Answer
The characteristic polynomial is $p(λ) = \text{det}(A - λI)$.
Answer for screen readers
The characteristic polynomial is given by $p(\lambda) = \text{det}(A - \lambda I)$.
Steps to Solve
- Define the Matrix A and Identity Matrix I
Given a square matrix $A$ of size $n \times n$, the identity matrix $I$ will also be of the same size. The identity matrix has 1's on the diagonal and 0's elsewhere.
- Set up the Expression A - λI
To find the characteristic polynomial, we need to subtract $λI$ from matrix $A$. This gives us the expression:
$$ B = A - λI $$
- Calculate the Determinant
Next, we need to find the determinant of the matrix $B$. This involves calculating the determinant of the matrix formed after subtraction:
$$ p(λ) = \text{det}(A - λI) $$
This determinant will be a polynomial in terms of $λ$.
- Expand the Determinant
Use cofactor expansion or a similar method to compute the determinant. This might involve finding minors and cofactors of the elements in the matrix $B$.
- Formulate the Characteristic Polynomial
The result of the determinant computation gives us the characteristic polynomial. This polynomial will typically be of degree $n$, where $n$ is the size of the matrix $A$.
The characteristic polynomial is given by $p(\lambda) = \text{det}(A - \lambda I)$.
More Information
The characteristic polynomial plays a vital role in linear algebra, as it provides information about the eigenvalues of the matrix $A$. Solving the characteristic polynomial allows us to find the eigenvalues, which have important applications in many fields, including stability analysis and quantum mechanics.
Tips
- Forgetting to subtract $λI$ correctly, leading to an incorrect expression for $A - λI$.
- Not correctly calculating the determinant, especially for larger matrices, which can lead to errors in the characteristic polynomial.
