Exercise · ZW-CHARACTERISTIC-POLYNOMIAL-008 · ★★☆ Intermediate
Find the characteristic polynomial of the rotation \(A=\begin{pmatrix}0&-1\\1&0\end{pmatrix}\).
Choose one option.
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- B. \(\lambda^2+1\)
Explanation
\(\det\begin{pmatrix}-\lambda&-1\\1&-\lambda\end{pmatrix}=\lambda^2+1\), which has no real roots: the rotation has no real eigenvalues.
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-003 The matrix A=[1 1; -2 4] has characteristic polynomial λ^2-5λ+6. What are its eigenvalues?
- 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-009 How many distinct eigenvalues does A=[2 1; 0 2] have?
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