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,

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).