Q.Find the value of the following: The number of all possible matrices of order $3 \times 3$ with each entry 0 or 1 is: (A) 27 (B) 18 (C) 81 (D) 512

Concept: Each entry in a $3 \times 3$ matrix is independent, and can be chosen in 2 ways (0 or 1).

Step 1: A $3 \times 3$ matrix has $3 \times 3 = 9$ entries.

Step 2: For each entry, there are 2 choices. By the fundamental principle of counting, total matrices = $2^9$. …

The key idea here is simple but powerful: a matrix is just a rectangular arrangement of numbers. When we talk about "all possible matrices" with entries restricted to 0 or 1, we are really counting the number of ways to fill each of the 9 positions with one of two choices.

This is a direct application of the fundamental principle of counting (also called the multiplication principle). If you have $n$ independent choices, each with $k$ possibilities, the total number of outcomes is $k^n$. There is no need for Inclusion-Exclusion here because the choices are independent — no restrictions overlap or exclude each other.

Let’s walk through it step by step.

1. Identify the number of entries.

   A $3 \times 3$ matrix has 3 rows and 3 columns. That gives $3 \times 3 = 9$ entries in total. Each entry is a separate "slot" to be filled.

2. Choices per entry.

   The problem says each entry can be either 0 or 1. That’s exactly 2 choices per entry.

3. Apply the multiplication principle.

   Since the choice for one entry does not affect the choice for another, we multiply the number of possibilities for each of the 9 entries:

   $2 \times 2 \times 2 \times \dots \text{(9 times)} = 2^9$ …

Method: Counting Matrices With Entries From a Finite Set

Use this to count how many matrices of a given order can be formed when each entry is chosen independently from a fixed set of values.

Steps

Step 1: Count the entries (positions to fill).

A matrix of order $m\times n$ has $m\times n$ positions. Each position is an independent slot to be filled.

Step 2: Count the choices per entry.

If each entry may be any of $k$ values (here $0$ or $1$, so $k=2$), every slot independently has $k$ options. …

Common Mistakes

Mistake 1: Miscounting the number of entries.

Why it's wrong: a $3\times3$ matrix has $3\times3=9$ entries, not $3+3=6$. Correct approach: multiply rows by columns to get the number of positions.

Mistake 2: Swapping base and exponent.

Why it's wrong: the count is (choices)$^{\text{entries}}=2^{9}$, not $9^{2}$ or $2\times9$. Correct approach: raise the number of choices per entry to the power of the number of entries. …