Question

The eigenvalues of the matrix
\[ \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix} \]
are _______ .

• A. 0, 2
• B. 2, 3
• C. 1, 3
• D. 1, 2

Hint:

To find the eigenvalues of a matrix, solve the characteristic equation \( {det}(A - \lambda I) = 0 \). The roots of the equation are the eigenvalues.

Answer 1

The Correct Option is C

To find the eigenvalues of a matrix, we solve the characteristic equation: \[ \det(A - \lambda I) = 0 \] where \( A \) is the matrix, \( \lambda \) is the eigenvalue, and \( I \) is the identity matrix. 
Step 1: Construct the matrix \( A - \lambda I \)
Given the matrix \( A = \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix} \), we subtract \( \lambda I \): \[ A - \lambda I = \begin{bmatrix} 1 - \lambda & 2 \\ 0 & 3 - \lambda \end{bmatrix} \] Step 2: Find the determinant of \( A - \lambda I \)
\[ \det(A - \lambda I) = (1 - \lambda)(3 - \lambda) - (0)(2) = (1 - \lambda)(3 - \lambda) \] Step 3: Solve the characteristic equation
\[ (1 - \lambda)(3 - \lambda) = 0 \Rightarrow \lambda = 1 \text{ or } \lambda = 3 \] Conclusion:
The eigenvalues of the matrix are 1 and 3.