Question

Find the number of non-zero determinants of order 2 with elements 0 or 1 only.

Answer 1

Hint: As it is given that the determinant is non zero so this means \[ad\ne bc\]. So we will take two cases to solve this problem. First case will be when ad is equal to 1 and bc is equal to 0. And the second case will be when ad is equal to 0 and bc is equal to 1. Now we will find the number of determinants in both these cases and add it to get our answer.

Complete step-by-step answer:
We will first write the general determinant,
\[\left| \left( \begin{matrix}
   a & b \\
   c & d \\
\end{matrix} \right) \right|=ad-bc........(1)\]
Now it is mentioned in the question that the determinant is non zero. So from equation (1) we get,
\[\Rightarrow ad-bc\ne 0......(2)\]
So from equation (2) we can say that \[ad\ne bc\].
Also it is given that a, b, c, d is either 0 or 1.
So our first case will be when ad is equal to 1 and bc is equal to 0.
Now for ad to be equal to 1 both a and d should be 1.
Similarly for bc to be equal to 0 we can have both b and c equal to 0 or b is 1 and c is 0 or b is 0 and c is 1.
So in this case from the above entries we can have \[\left| \left( \begin{matrix}
   1 & 0 \\
   0 & 1 \\
\end{matrix} \right) \right|\], \[\left| \left( \begin{matrix}
   1 & 1 \\
   0 & 1 \\
\end{matrix} \right) \right|\], \[\left| \left( \begin{matrix}
   1 & 0 \\
   1 & 1 \\
\end{matrix} \right) \right|\]. So we are having 3 non zero determinants in this case.
Now our second case will be when ad is equal to 0 and bc is equal to 1.
Now for bc to be equal to 1 both b and c should be 1.
Similarly for ad to be equal to 0 we can have both a and d equal to 0 or a is 1 and d is 0 or a is 0 and d is 1.
So in this case from the above entries we can have \[\left| \left( \begin{matrix}
   0 & 1 \\
   1 & 0 \\
\end{matrix} \right) \right|\], \[\left| \left( \begin{matrix}
   1 & 1 \\
   1 & 0 \\
\end{matrix} \right) \right|\], \[\left| \left( \begin{matrix}
   0 & 1 \\
   1 & 1 \\
\end{matrix} \right) \right|\]. So we are having total 3 non zero determinants in this case too.
Thus the total number of non-zero determinants of order 2 with elements 0 or 1 only is 6.

Note: Remembering the formula of the determinant \[\left| A \right|=ad-bc\] of a matrix is the key here. Also order 2 means that it has two rows and two columns. The determinant is a scalar value that can be computed from the elements of a square matrix A and it is denoted by det(A).