📘 Question
Write the number of all possible matrices of order \(2 \times 2\) with each entry being 1, 2, or 3.
✏️ Step-by-Step Solution
A \(2 \times 2\) matrix has 4 entries:
\[
\begin{bmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{bmatrix}
\]
Each entry can be filled with any of the numbers: \(1, 2, 3\).
Step 1: Count choices per entry
Each position has 3 choices.
Step 2: Total number of matrices
\[
\text{Total} = 3^4
\]
\[
= 81
\]
✅ Final Answer
\[
\boxed{81}
\]
💡 Key Concept
If a matrix has \(m \times n\) entries and each entry has \(k\) choices, then:
\[
\text{Total matrices} = k^{m \times n}
\]
Here: \(k = 3\), \(m \times n = 4\), so total = \(3^4 = 81\).