Total number of possible matrices of order 3 × 3 with each entry 2 or 0 is

The correct option is 512.

Let us break down the solution step-by-step to understand why this is the correct answer:

Step 1: Understand the structure of the matrix
A matrix of order 3 × 3 has 3 rows and 3 columns.
The total number of elements (or positions) in a 3 × 3 matrix is given by:

$$3\times 3=9$$

So, there are 9 distinct positions in the matrix that need to be filled.

Step 2: Identify the choices for each position
According to the problem statement, each entry in the matrix can be either 2 or 0.
This means that for every single position in the matrix, we have exactly 2 choices (either the number 2 or the number 0).

Step 3: Calculate the total number of possible matrices
Since the choice of entry for any position is independent of the choices made for the other positions, we can use the fundamental multiplication principle of counting.
For 9 positions, where each position has 2 choices, the total number of possible matrices is:

$$2^9$$

Let us calculate the value of 2⁹:

$$2^9=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2=512$$

Therefore, the total number of all possible matrices of order 3 × 3 with each entry 2 or 0 is 512.