Let M be the set of all 2 2 matrices with entries from the set R of real numbers. Then the function f: M R defined by f(A)=|A| for every A M is

Step-by-step solution

The function f: M R is defined as f(A) = |A| , where M is the set of all 2 2 matrices with real entries and |A| denotes the determinant of matrix A . To check if f is one-one, consider two distinct matrices A = pmatrix 1 & 0 0 & 1 pmatrix and B = pmatrix 2 & 0 0 & 0.5 pmatrix . Both matrices are in M and A B . The determinant of A is |A| = (1)(1) - (0)(0) = 1 . The determinant of B is |B| = (2)(0.5) - (0)(0) = 1 . Since f(A) = f(B) but A B , the function f is not one-one. To check if f is onto, consider any real nu