Dice Matrix Probability: Distinct & Non-Singular

The problem requires finding the probability that a $2 \times 2$ matrix formed by rolling four dice simultaneously contains all distinct entries and is non-singular.

Total Outcomes Calculation

Each of the four dice can land on any number from 1 to 6. Therefore, the total number of possible outcomes when rolling four dice is:

Total Outcomes = $6 \times 6 \times 6 \times 6 = 6^4 = 1296$

Favorable Outcomes: Distinct Entries & Non-Singular Matrix

A favorable outcome requires the $2 \times 2$ matrix, say $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, to satisfy two conditions:

1. All entries ($a, b, c, d$) are distinct numbers chosen from $\{1, 2, 3, 4, 5, 6\}$.
2. The matrix is non-singular, meaning its determinant ($ad - bc$) is not equal to zero ($ad \neq bc$).

Step 1: Count Matrices with Distinct Entries

The number of ways to choose 4 distinct numbers from the 6 possible outcomes and arrange them in the 4 positions of the matrix is calculated using permutations $P(n, k) = \frac{n!}{(n-k)!}$.

Number of matrices with distinct entries = $P(6, 4) = \frac{6!}{(6-4)!} = \frac{6!}{2!} = 6 \times 5 \times 4 \times 3 = 360$

Step 2: Count Singular Matrices with Distinct Entries

We need to identify matrices where $a, b, c, d$ are distinct, and $ad = bc$. This condition implies that the four distinct numbers must be partitionable into two pairs with equal products.

The sets of 4 distinct numbers from $\{1, ..., 6\}$ that satisfy this are:

• Set {1, 2, 3, 6}: Pairs {1, 6} and {2, 3}. Product = 6.
• Set {2, 3, 4, 6}: Pairs {2, 6} and {3, 4}. Product = 12.

For each set, we can form singular matrices. Consider the set {1, 2, 3, 6}:

• Case 1: Assign $\{a, d\}$ from $\{1, 6\}$ (2 ways: (1,6) or (6,1)) and $\{b, c\}$ from $\{2, 3\}$ (2 ways: (2,3) or (3,2)). This gives $2 \times 2 = 4$ matrices.
• Case 2: Assign $\{a, d\}$ from $\{2, 3\}$ (2 ways) and $\{b, c\}$ from $\{1, 6\}$ (2 ways). This gives $2 \times 2 = 4$ matrices.
• Total singular matrices for this set = $4 + 4 = 8$.

Similarly, the set {2, 3, 4, 6} also yields 8 singular matrices.

Total singular matrices with distinct entries = $8 + 8 = 16$.

Step 3: Count Non-Singular Matrices with Distinct Entries

Subtract the count of singular matrices with distinct entries from the total count of matrices with distinct entries.

Number of non-singular distinct matrices = $360 - 16 = 344$

Probability Calculation

The probability is the ratio of the number of favorable outcomes (non-singular matrices with distinct entries) to the total possible outcomes.

Probability = $\frac{\text{Number of non-singular distinct matrices}}{\text{Total possible outcomes}}$

Probability = $\frac{344}{1296}$

Simplify the fraction:

$\frac{344 \div 8}{1296 \div 8} = \frac{43}{162}$

Conclusion

The probability that the formed matrices have all different entries and are non-singular is $\frac{43}{162}$.