📘 Question
If \(A = [a_{ij}]\) is a scalar matrix of order \(n \times n\) such that \(a_{ii} = k\), then find \(\text{Trace}(A)\).
(a) \(nk\)
(b) \(n + k\)
(c) \(\frac{n}{k}\)
(d) none of these
✏️ Step-by-Step Solution
Step 1: Understand scalar matrix
All diagonal elements are equal:
\[
a_{11} = a_{22} = \cdots = a_{nn} = k
\]
Step 2: Use trace formula
\[
\text{Trace}(A) = a_{11} + a_{22} + \cdots + a_{nn}
\]
Step 3: Add values
\[
\text{Trace}(A) = k + k + \cdots + k \; (n \text{ times}) = nk
\]
✅ Final Answer
\[
\boxed{(a)\; nk}
\]
💡 Key Concept
Trace of a scalar matrix = (number of diagonal elements) × (common value).