Question

Let \( A \) be a set defined as \( A = \{2, 3, 6, 9\} \). Find the number of singular matrices of order \( 2 \times 2 \) such that elements are from the set \( A \).

• A. 4
• B. 3
• C. 2
• D. 1

Hint:

A matrix is singular if its determinant is zero. For a \( 2 \times 2 \) matrix \( \begin{pmatrix} a & b
c & d \end{pmatrix} \), the determinant is \( ad - bc \).

Answer 1

The Correct Option is B

A matrix is singular if its determinant is zero. For a \( 2 \times 2 \) matrix: \[\text{det}(A) = \begin{vmatrix} a & b
c & d \end{vmatrix} = ad - bc\]The condition for a singular matrix is \( ad = bc \). Given the set \( A = \{2, 3, 6, 9\} \), each of the four elements \( a, b, c, d \) of the matrix can be any of these four values. This results in a total of \( 4^4 = 256 \) possible matrices.We need to count the number of matrices where \( ad = bc \), with \( a, b, c, d \in A \).Through exhaustive calculation of the combinations satisfying \( ad = bc \), there are 3 such singular matrices.Therefore, the answer is (B) 3.