Determinants

To solve the problem, we need to analyze the function defined by , where and is the set of all 2x2 matrices of the form:

1. Understanding the Function:
The function maps each 2x2 real matrix to a real number which is the determinant of the matrix:

2. Determine If the Function Is Onto:
We want to see if for every real number , there exists a matrix such that .
Take a matrix . Then, .
Hence, any real number can be obtained as the determinant of some matrix. So, is onto.

3. Determine If the Function Is One-One:
If were one-one, then different matrices would have different determinants.
But consider matrices:
and .
Both have , but .
So, is not one-one.

Final Answer:
The function is onto but not one-one.