📘 Question
If
\[
[2 \;\; 1 \;\; 3]
\begin{bmatrix}
-1 & 0 & -1 \\
-1 & 1 & 0 \\
0 & 1 & 1
\end{bmatrix}
\begin{bmatrix}
1 \\
0 \\
-1
\end{bmatrix}
= A
\]
Find the order of matrix \(A\).
✏️ Step-by-Step Solution
Step 1: Identify dimensions
- \([2 \;\; 1 \;\; 3]\) → \(1 \times 3\) matrix
- Middle matrix → \(3 \times 3\)
- Column matrix → \(3 \times 1\)
Step 2: Multiply step-by-step
First multiplication:
\[
(1 \times 3)(3 \times 3) = 1 \times 3
\]
Second multiplication:
\[
(1 \times 3)(3 \times 1) = 1 \times 1
\]
Step 3: Final order
Thus, matrix \(A\) is:
\[
1 \times 1
\]
✅ Final Answer
\[
\boxed{1 \times 1}
\]
💡 Key Concept
If \(A_{m \times n} \times B_{n \times p} = C_{m \times p}\), then the resulting matrix has order \(m \times p\).