Q.Total number of possible matrices of order $3 \times 3$ with each entry $2$ or $0$ is
(A) $9$
(B) $27$
(C) $81$
(D) $512$

Concept: Matrix Order Possibilities — For each of the $3 \times 3 = 9$ entries, there are exactly 2 choices ($2$ or $0$).

Step 1: Total entries in a $3 \times 3$ matrix = $3 \times 3 = 9$.

Step 2: Each entry can be filled independently in 2 ways. …

The key insight here is that a matrix is defined entirely by its entries. For a $3 \times 3$ matrix, there are exactly 9 positions to fill. The problem says each entry must be either 2 or 0 — that’s two choices per position, with no other restrictions.

1. Count the positions.

   A $3 \times 3$ matrix has 3 rows and 3 columns, so $3 \times 3 = 9$ entries. Each entry is independent of the others.

2. Choices per entry.

   For each of the 9 positions, you can put either a 2 or a 0. That’s 2 possibilities per slot.

3. Apply the multiplication principle.

   When choices are independent, the total number of combinations is the product of the number of choices for each slot.

4. Compute $2^9$. $2^9 = 512$ …

Method: Counting matrices by the multiplication principle

To count how many matrices of a given order can be formed when each entry is chosen from a fixed set of values, count the entries and raise the number of choices per entry to that power.

Steps

Step 1: Count the total number of entries.

An $m\times n$ matrix has $m\times n$ entries; a $3\times3$ matrix has $9$.

Step 2: Count the choices for a single entry.

Here each entry can be one of two values ($2$ or $0$), so there are $2$ choices per entry. …

Common Mistakes

Mistake 1: Answering $9$ (just the number of entries).

Why it's wrong: $9$ counts positions, not the number of distinct matrices; each position can independently be $2$ or $0$. Correct approach: raise the choices per entry to the number of entries: $2^9$.

Mistake 2: Computing $2\times9=18$ instead of $2^{9}$.

Why it's wrong: independent choices multiply as a power, not a product of the two counts. Correct approach: use $2^{9}=512$. …