Question

The number of all possible matrices of order \[3 \times 3\] with each entry 0 or 1 is
(a) 9
(b) 18
(c) 27
(d) 512

Answer 1

Hint: To solve the question given above, we will first find out what is a matrix. Then we will check what is the number of choices available to each element of the matrix. Then we will multiply the number of choices of every element to find the total number of all matrices.

Complete step by step solution:
Before we solve the question, we must know what a matrix is. A matrix is a rectangular array of numbers, symbols or expressions arranged in rows and columns. In the question, we are given a \[3 \times 3\] matrix which means it has 3 rows and 3 columns. Thus, the total number of elements in the matrix will be \[ = 3 \times 3 = 9\].
\[A = {\left( {\begin{array}{*{20}{c}}{{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\{{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\{{a_{31}}}&{{a_{32}}}&{{a_{33}}}\end{array}} \right)_{3 \times 3}}\]
Let A be the \[3 \times 3\] matrix as shown above. Now we will find out the total number of choices each element of the matrix has. We will now check for \[{a_{11}}\].
Here, we can see that \[{a_{11}}\] can either be 0 or 1. Thus, the total choices available for \[{a_{11}}\] are 2. Now, we will check for \[{a_{12}}\] . Here, \[{a_{12}}\] can have the value either 1 or 0. Thus, the total number of choices available to \[{a_{12}}\] are 2. Similarly, for \[{a_{13}}\] , the total choices available will be 2. Thus, all the other elements have a total number of choices equal to 2. Thus, the total number of matrices will be equal to the product of the choices available to each element. Thus, we have:
Total matrices \[ = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\]
Total matrices \[ = 512\]
Hence, option (d) is correct.

Note: We can also create the general formula for this type of questions where we know the total number of elements and total number of choices available. This is given by:
Number of matrices \[ = {a^{\left( {m \times n} \right)}}\]
Where, a is the total number of choices available, m is the number of rows and n is the number of columns.
On putting the values of \[m = 3\,,\,n = 3{\text{ and }}a = 2\] we will get:
Total matrices \[ = {\left( 2 \right)^{3 \times 3}}\]
 Total matrices \[ = {2^9}\]
Total matrices \[ = 512\]