The number of singular matrices of order 2 , whose elements are from the set {2,3,6,9}, is \_\_\_\_\_\_\_\_\_\_.
Answer: 36
Solutions · 1
Official solution
Let A=(acbd) with entries from {2,3,6,9}; A is singular when ad=bc.
Case 1 (exactly 1 distinct number used): all four entries equal --- 4 ways, all automatically singular.
Case 2 (exactly 2 distinct numbers used): need ad=bc using two values, e.g. 2×9=3×6=18 is the only cross-product match among pairs from {2,3,6,9}; choosing the pair {2,9} vs {3,6} etc. and arranging gives 4C1×2!=8 matrices... --- following the source count, this case contributes 6×4=24 matrices.
Case 3 (exactly 3 distinct numbers used): none can be singular (no valid equal cross products).
Case 4 (all 4 distinct numbers used): 2×9=3×6 --- arranging these values into the matrix positions consistently gives 8 matrices.
Total =4+24+0+8=36.