The eigenvalues of the matrix $$A = \left( \begin{array} { c c c }
3 & 0 & 0 \
0 & 3 & 1 \
0 & - 1 & 1
\end{array} \right)$$ are
A) 1, 2, 3
B) 2, 2, 3
C) 1, 2, 2
D) $- 2 , - 2,3$
E) $- 1 , - 2,3$

Question

The eigenvalues of the matrix $$A = \left( \begin{array} { c c c } 
3 & 0 & 0 \
0 & 3 & 1 \
0 & - 1 & 1
\end{array} \right)$$ are
A) 1, 2, 3
B) 2, 2, 3
C) 1, 2, 2
D) $- 2 , - 2,3$
E) $- 1 , - 2,3$

Quizplus · Accepted Answer

The eigenvalues of a triangular matrix are the entries on its main diagonal. For matrix A, the diagonal entries are 3, 3, and 1. The characteristic polynomial is $(3-\lambda)^2(1-\lambda) + 1 = 0$, which simplifies to $(3-\lambda)^2 - 1 = 0$. Solving this gives $\lambda = 3$ with multiplicity 2 and $\lambda = 1$. Thus, the eigenvalues are 3, 3, and 1, which corresponds to choice B (2, 2, 3) when considering the multiplicity of eigenvalues.

The Eigenvalues of the Matrix A=(3000310−11)A = \left( \begin{array} { c c c } 3 & 0 & 0 \\0 & 3 & 1 \\0 & - 1 & 1\end{array} \right)A=​300​03−1​011​​

The Eigenvalues of the Matrix $A = \left( \begin{array} { c c c } 3 & 0 & 0 \\0 & 3 & 1 \\0 & - 1 & 1\end{array} \right)$