📘 Question
If
\[
A =
\begin{bmatrix}
n & 0 & 0 \\
0 & n & 0 \\
0 & 0 & n
\end{bmatrix}
,\quad
B =
\begin{bmatrix}
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{bmatrix}
\]
Find \(AB\).
✏️ Step-by-Step Solution
Step 1: Recognize the matrix
Matrix \(A\) is a scalar matrix:
\[
A = nI
\]
Step 2: Use property
Multiplying a scalar matrix:
\[
AB = nI \cdot B = nB
\]
Step 3: Multiply each element
\[
AB =
\begin{bmatrix}
na_1 & na_2 & na_3 \\
nb_1 & nb_2 & nb_3 \\
nc_1 & nc_2 & nc_3
\end{bmatrix}
\]
✅ Final Answer
\[
\boxed{
\begin{bmatrix}
na_1 & na_2 & na_3 \\
nb_1 & nb_2 & nb_3 \\
nc_1 & nc_2 & nc_3
\end{bmatrix}
}
\]
💡 Key Concept
A scalar matrix behaves like a number. So multiplying \(nI\) with any matrix simply multiplies each element by \(n\).