Question

Find the number of all possible matrices of order $ 3 \times 3 $ with each entry $ 0 $ or $ 1 $ ?

Answer 1

Hint: In the given question, we are required to find the total number of possible matrices of order $ 3 \times 3 $ whose all entries are either $ 0 $ or $ 1 $ . So, there are only two possible values for each entry of the $ 3 \times 3 $ matrix. There are a total of $ 9 $ entries in a $ 3 \times 3 $ matrix and we need to fill all the nine entries with either $ 0 $ or $ 1 $.

Complete step by step solution:
The number of elements in a matrix of order $ 3 \times 3 $ are $ 9 $ .
All the nine elements have to be either $ 0 $ or $ 1 $ .
So, Number of entries is $ 2 $ .
Thus each of the nine elements can be given by either $ 0 $ or $ 1 $ .
That is, each of the nine elements can be filled in two possible ways. So, we multiply the number of ways of filling up each element spot in the required matrix as the events are happening one after the other. This is called the fundamental principle of counting.
Therefore, the required number of possible matrices of order $ 3 \times 3 $ $ = {2^9} = 512 $
Hence, the number of all possible matrices of order $ 3 \times 3 $ with each entry $ 0 $ or $ 1 $ is $ 512 $ .
So, the correct answer is “512”.

Note: These type of problems which require us to find the number of matrices that can be formed following a certain set of conditions require us to have a thorough knowledge of permutations and combinations. We can find that number of possible matrices by applying the fundamental theorem of counting.