Question

when the eigenvalues of

\\a = \

$$\begin{bmatrix} a & b \\\\ 0 & d \\end{bmatrix}$$

\\

are 0 and 1, what are the possible values of a and d?

a = 0 and d = -1
a = -1 and d = 0
a = 0 and d = 1
a = 1 and d = 0

Explanation:

Identify matrix structure

The given matrix is:

$$ A =
LATEXBLOCK0
$$

This is an upper triangular matrix.

Determine eigenvalues

For any triangular matrix, the eigenvalues are the entries on its main diagonal.
Therefore, the eigenvalues of \(A\) are \(a\) and \(d\).

Match given eigenvalues

We are given that the eigenvalues of \(A\) are \(0\) and \(1\).
Thus, the set of diagonal entries \(\{a, d\}\) must be equal to the set of eigenvalues \(\{0, 1\}\).
This yields two possible cases:

1. \(a = 0\) and \(d = 1\)
2. \(a = 1\) and \(d = 0\)

Evaluate the options

Let's check the given multiple-choice options:

• \(a = 0\) and \(d = -1\) (Incorrect)
• \(a = -1\) and \(d = 0\) (Incorrect)
• \(a = 0\) and \(d = 1\) (Correct)
• \(a = 1\) and \(d = 0\) (Correct)

Among the choices, both \(a = 0 \text{ and } d = 1\) and \(a = 1 \text{ and } d = 0\) are mathematically correct. Looking at the options, we select the matching choices.

Answer:

• a = 0 and d = -1
• a = -1 and d = 0
• a = 0 and d = 1 (Correct answer)
• a = 1 and d = 0 (Correct answer)