Four dice are thrown simultaneously and the numbers shown on these dice are recorded in $$2 \times 2$$ matrices. The probability that such formed matrices have all different entries and are non-singular, is:

Four dice are thrown and the four numbers are arranged in a $$2 \times 2$$ matrix $$M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$. Since each die shows a value in $$\{1, 2, 3, 4, 5, 6\}$$ and the four positions in the matrix are ordered, the total number of outcomes is $$6^4 = 1296$$.

We need all four entries to be different and $$\det(M) = ad - bc \neq 0$$. The number of ordered arrangements of 4 distinct values chosen from 6 is $$P(6, 4) = 6 \times 5 \times 4 \times 3 = 360$$ (we pick 4 values from 6 in order, since positions $$a, b, c, d$$ are distinguishable).

Now we count how many of these 360 matrices are singular ($$ad = bc$$). We need two disjoint unordered pairs from $$\{1, \ldots, 6\}$$ with the same product. Listing all products of pairs of distinct elements: $$1 \times 2 = 2$$, $$1 \times 3 = 3$$, $$1 \times 4 = 4$$, $$1 \times 5 = 5$$, $$1 \times 6 = 6$$, $$2 \times 3 = 6$$, $$2 \times 4 = 8$$, $$2 \times 5 = 10$$, $$2 \times 6 = 12$$, $$3 \times 4 = 12$$, $$3 \times 5 = 15$$, $$3 \times 6 = 18$$, $$4 \times 5 = 20$$, $$4 \times 6 = 24$$, $$5 \times 6 = 30$$.

The only products that appear for two disjoint pairs are: product $$6$$ from $$\{1, 6\}$$ and $$\{2, 3\}$$, and product $$12$$ from $$\{2, 6\}$$ and $$\{3, 4\}$$. (Other repeated products, if any, would share a common element.)

For each such pair of disjoint pairs, we assign one pair to positions $$(a, d)$$ and the other to $$(b, c)$$: there are 2 ways to decide which pair goes to $$(a, d)$$. Each pair can be placed in its two positions in $$2$$ ways (e.g., $$(a, d) = (1, 6)$$ or $$(6, 1)$$). So each product class gives $$2 \times 2 \times 2 = 8$$ singular matrices.

Total singular matrices with all distinct entries: $$8 + 8 = 16$$. Hence the number of non-singular matrices with all distinct entries is $$360 - 16 = 344$$.

The required probability is $$\frac{344}{1296}$$. Simplify by dividing numerator and denominator by 8: $$\frac{43}{162}$$.