Question

A determinant $\Delta$ is chosen at random from the set of all determinant of order two with elements 0 and 1 only .Then the probability with the determinant

Answer 1

Considering element 1 and 0, there are 16 ways of constructing a determinant.
Out of which only 6 determinants give non-zero answers.
Hence the probability of having a non zero $2\times 2$ determinant is
$=\frac{6}{16}$
$=\frac{3}{8}$.
Hence the probability of having a null determinant will be
$=1-\frac{3}{8}$
$=\frac{5}{8}$
Now if we take only 0 and 1's to construct a determinant then the determinant value will fall in the following set
$\{-1,0,1\}$
Hence the probability that $△=2$ is 0.
Now the no of determinants with $△=1$ is equal to the determinants with $△=-1$ due to symmetric nature.
Hence
Probability of having determinant with $△=1$
=Probability of having determinant with $△=-1$
$=\left(\frac{1}{2}\right)$the probability of having a non zero $2\times 2$ determinant
$=\frac{1}{2}.\frac{3}{8}$
$=\frac{3}{16}$