A $$2 \times 2$$ matrix is filled with four distinct integers randomly chosen from the set {1,2,3,4,5,6}.
Then the probability that the matrix generated in such a way is singular is

Let the matrix be $$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$$, where a, b, c, and d are all distinct.

The number of matrices formed using numbers from set {1,2,3,4,5,6} $$=6\times 5\times 4\times 3=360$$

Now, for the number of singular matrices, we should have the determinant as 0. This means that $$ad-bc=0$$ => $$ad=bc$$. Total number of sets of {a,b,c,d} which satisfy this inequality is -

$$a=1,\ d=6\ and\ c=2,\ b=3$$ -> 2 ways [ when a and c are swapped, and b and d are swapped]
$$a=1,\ d=6\ and\ c=3,\ b=2$$ -> 2 ways [ when a and c are swapped, and b and d are swapped]
$$a=6,\ d=1\ and\ c=2,\ b=3$$ -> 2 ways [ when a and c are swapped, and b and d are swapped]
$$a=6,\ d=1\ and\ c=3,\ b=2$$ -> 2 ways [ when a and c are swapped, and b and d are swapped]
$$a=2,\ d=6\ and\ c=3,\ b=4$$ -> 2 ways [ when a and c are swapped, and b and d are swapped]
$$a=2,\ d=6\ and\ c=4,\ b=3$$ -> 2 ways [ when a and c are swapped, and b and d are swapped]
$$a=6,\ d=2\ and\ c=3,\ b=4$$ -> 2 ways [ when a and c are swapped, and b and d are swapped]
$$a=6,\ d=2\ and\ c=4,\ b=3$$ -> 2 ways [ when a and c are swapped, and b and d are swapped]

Thus, there are 16 cases possible when the matrix will be singular.

Probability = $$\dfrac{16}{360}=\dfrac{2}{45}$$