Step 1 : Concept:
This matching question involves computing the trace, determinant, characteristic polynomial, and eigenvalues of a $2 \times 2$ matrix. Step 2 : Key Formulas and Approach:
For $A = \begin{bmatrix} a & b c & d \end{bmatrix}$:
1. $\text{Trace}(A) = a + d$
2. $\det(A) = ad - bc$
3. Characteristic polynomial: $p(\lambda) = \det(\lambda I - A) = \lambda^2 - \text{Trace}(A)\lambda + \det(A)$
4. Eigenvalues of a triangular matrix are its diagonal entries. Step 3 : Step-by-step Explanation:
• Item A:
$\text{Trace}(A) = \text{Sum of diagonal elements} = 1 + 2 = 3$. Matches with III.
• Item B:
$\det(A) = (1)(2) - (0)(5) = 2$. Matches with I.
• Item C:
$\det(\lambda I - A) = \lambda^2 - \text{Trace}(A)\lambda + \det(A) = \lambda^2 - 3\lambda + 2$. Matches with IV.
• Item D:
Since $A = \begin{bmatrix} 1 & 0 5 & 2 \end{bmatrix}$ is lower triangular, its eigenvalues are the diagonal entries $1$ and $2$. Matches with II. Step 4 : Final Answer:
The correct matching is A-III, B-I, C-IV, D-II, which corresponds to option (B).
