Let M = [a_ij] be a 2x2 matrix whose four entries are all distinct and each entry belongs to the set {1, 2, 5, 10}. Find the number of such matrices M that are singular.

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0. The four entries are a permutation of all elements of {1,2,5,10} (since all 4 are used, all distinct). We need a11*a22 = a12*a21. Products of pairs from {1,2,5,10}: 1*2=2, 1*5=5, 1*10=10, 2*5=10, 2*10=20, 5*10=50. For two diagonals to have equal products: need two distinct pairs with the same product. From the list: 1*10=10 and 2*5=10. So the pairs {1,10} and {2,5} each have product 10. For the matrix to be singular with these four elements: diagonal pair (a11,a22) and anti-diagonal pair (a12,a21) must have the same product. The only option is {1,10} on one diagonal and {2,5} on the other (product 10 = 10). Number of such arrangements: assign {1,10} to main diagonal: (1,10) or (10,1) -> 2

Let M = [a_ij] be a 2x2 matrix whose four entries are all distinct and each entry belongs to the set {1, 2, 5, 10}. Find the number of such matrices M that are singular.

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