Probability of a singular matrix using the first ten prim…

Identify the total possible matrices

A

matrix has four entries. If each entry is chosen from a set of 10 prime numbers, the total number of unique matrices is:

2. Set the condition for singularity

A matrix

is singular if its determinant is zero:

Since all entries are prime numbers, we must find all pairs

and

such that their products are equal.

3. Categorize equal products

Because the entries are prime, the product

can only equal

in two specific scenarios:

Case 1: All entries are the same (

)

There are 10 such matrices (one for each prime in the set).

Case 2: Entries are same in pairs

Subcase A:

and

, but

.

For the first pair

, we have 10 choices. For the second pair

, we have 9 remaining choices.

matrices.

Subcase B:

and

, but

.

Similarly, there are

matrices.

4. Calculate total singular matrices

Summing the cases where

:

5. Final Probability

The probability

is the ratio of singular matrices to the total:

Correct Answer

Based on the calculation, the correct option is (c) 19/(10^3).

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