The number of singular matrices of order 2, whose elements are from the set $ \{2, 3, 6, 9\} $ is:
Show 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.
Consider a \(2 \times 2\) matrix with the general form \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \). This matrix is singular if its determinant is zero, which means \( ad - bc = 0 \), or equivalently, \( ad = bc \). Each element \( a, b, c, d \) is selected from the set \( \{2, 3, 6, 9\} \), which contains 4 distinct elements. The total number of possible \(2 \times 2\) matrices that can be constructed is \( 4^4 = 256 \). We determine the count of these matrices that satisfy the condition \( ad = bc \) by examining all possible 4-tuples \( (a, b, c, d) \in \{2, 3, 6, 9\}^4 \) and identifying those for which \( ad = bc \). Through exhaustive enumeration or programmatic checking, it is found that the number of singular matrices is 36.
