Discussion
For the matrix \(M = \begin{bmatrix} 1 & 4 & 5 \\ 0 & 2 & 6 \\ 0 & 0 & 3 \end{bmatrix}\), consider the following statements:
P: 3 is an eigenvalue of M.
Q: \(\begin{bmatrix} 4 \\ 1 \\ 0 \end{bmatrix}\) is an eigenvector of M.
R: \(\begin{bmatrix} 4 \\ 2 \\ 0 \end{bmatrix}\) is an eigenvector of M.
Which of the above statements are TRUE?
Answer: A
P and Q are true and R is not, so the answer is P and Q, but not R.
M is upper triangular, so its eigenvalues are its diagonal entries 1, 2 and 3; P is true. Test the vectors:
\[ \begin{aligned} M\begin{bmatrix} 4 \\ 1 \\ 0 \end{bmatrix} &= \begin{bmatrix} 4 + 4 \\ 2 \\ 0 \end{bmatrix} = 2\begin{bmatrix} 4 \\ 1 \\ 0 \end{bmatrix} \\ M\begin{bmatrix} 4 \\ 2 \\ 0 \end{bmatrix} &= \begin{bmatrix} 12 \\ 4 \\ 0 \end{bmatrix} \end{aligned} \]
The first is twice the vector, so it is an eigenvector for eigenvalue 2 (Q true). (12, 4, 0) is not a multiple of (4, 2, 0), since 12/4 = 3 but 4/2 = 2, so R is false. Every option that includes R or excludes P is therefore wrong.
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