1.
Let Mat2(R) be the set of all 2 2 matrices having entries in the set R of real numbers. Let f :
Mat2(R) ? R be the ?determinant function? given by (a) Prove or disprove: f is injective.
Solution
The determinant function is injective if different values of a,b,c,d always give different values of
a*d-b*c; or, in other words, there aren't two different matrices that have the same determinant. It
is NOT injective. As a counterexample, consider the two matrices: [4 0] [0 1] and [2 0] [0 2] The
determinant of the first one is 4*1-0*0 = 4. The determinant of the second one is 2*2-0*0=4.
Since different 2x2 matrices can have different determinants, the function is NOT injective.