Write the number of all possible matrices of order 2 × 2 with each entry 1, 2 or 3.

We are given with the information that,

Each element of the 2 × 2 matrix can be filled in 3 ways, either 1, 2 or 3.

We need to find the number of total 2 × 2 matrices with each entry 1, 2 or 3.

Let A be 2 × 2 matrix such that,

Note that, there are 4 elements in the matrix.

So, if 1 element can be filled in 3 ways, either 1, 2 or 3.

That is,

Number of ways in which 1 element can be filled = 3¹

Then,

Number of ways in which 4 elements can be filled = 3⁴

⇒ Number of ways in which 4 elements can be filled = 81

Thus, total number of 2 × 2 matrices with each entry 1, 2 or 3 is 81.