📘 Question
If \(S = [s_{ij}]\) is a scalar matrix such that \(s_{ii} = k\), and \(A\) is a square matrix of the same order, then:
\[
AS = SA = \; ?
\]
(a) \(Ak\)
(b) \(k + A\)
(c) \(kA\)
(d) \(kS\)
✏️ Step-by-Step Solution
Step 1: Understand scalar matrix
A scalar matrix is of the form:
\[
S = kI
\]
Step 2: Multiply with matrix \(A\)
\[
AS = A(kI) = k(AI) = kA
\]
\[
SA = (kI)A = k(IA) = kA
\]
Step 3: Final result
\[
AS = SA = kA
\]
✅ Final Answer
\[
\boxed{(c)\; kA}
\]
💡 Key Concept
A scalar matrix behaves like a number. So multiplying it with any matrix is equivalent to scalar multiplication.