Question

The number of singular matrices of order 2, whose elements are from the set $ \{2, 3, 6, 9\} $ is:

Hint:

To find the number of singular matrices, remember that the determinant condition \( ad = bc \) must hold for each combination of matrix elements. This simplifies to finding matching pairs of products from the given set.

Answer 1

Correct Answer: 36

Let the general form of a \(2 \times 2\) matrix be: \[ \begin{bmatrix} a & b c & d \end{bmatrix} \] The matrix is singular if its determinant is zero: \[ \det = ad - bc = 0 \Rightarrow ad = bc \] Each entry \( a, b, c, d \) is chosen from the set \( \{2, 3, 6, 9\} \), which has 4 elements. 
The total number of \(2 \times 2\) matrices that can be formed is: \[ 4^4 = 256 \] We now count how many of these satisfy \( ad = bc \). 
We do this by checking all possible 4-tuples \( (a, b, c, d) \in \{2, 3, 6, 9\}^4 \), and count those for which \( ad = bc \). 
Using brute-force checking (e.g., via code or enumeration), we find that: \[ \text{Number of singular matrices} = 36 \]