Exercise · ZW-CHARACTERISTIC-POLYNOMIAL-003 · ★☆☆ Basic
The matrix \(A=\begin{pmatrix}1&1\\-2&4\end{pmatrix}\) has characteristic polynomial \(\lambda^2-5\lambda+6\). What are its eigenvalues?
Choose one option.
Show answer
- C. \(2\) and \(3\)
Explanation
The eigenvalues are the roots: \(\lambda^2-5\lambda+6=(\lambda-2)(\lambda-3)=0\), that is, 2 and 3.
Review the concept: Characteristic polynomial
The characteristic polynomial \(p_A\) of a square matrix \(A\) is the determinant of the matrix minus a multiple \(\lambda I\) of the identity \(I\), viewed as a polynomial in the variable \(\lambda\).
\[p_A(\lambda)=\det(A-\lambda I)\]
More practice problems: characteristic polynomial
- ZW-CHARACTERISTIC-POLYNOMIAL-001 Compute p_A(λ)=det(A-λ I) for A=[2 1; 1 2].
- ZW-CHARACTERISTIC-POLYNOMIAL-002 What is the characteristic polynomial of A=[3 0; 0 5]?
- ZW-CHARACTERISTIC-POLYNOMIAL-004 What is the degree of the characteristic polynomial of a 4×4 matrix?
- ZW-CHARACTERISTIC-POLYNOMIAL-005 Compute the characteristic polynomial of A=[1 2; 3 4].
- ZW-CHARACTERISTIC-POLYNOMIAL-006 For A=[4 1; 3 2], what is the constant term of p_A(λ)?
- ZW-CHARACTERISTIC-POLYNOMIAL-007 What are the eigenvalues of A=[2 1 4; 0 3 5; 0 0 -1]?
- ZW-CHARACTERISTIC-POLYNOMIAL-008 Find the characteristic polynomial of the rotation A=[0 -1; 1 0].
- ZW-CHARACTERISTIC-POLYNOMIAL-009 How many distinct eigenvalues does A=[2 1; 0 2] have?
← ZW-CHARACTERISTIC-POLYNOMIAL-002 ZW-CHARACTERISTIC-POLYNOMIAL-004 →