Question

Suppose that the trace of a 2x2 matrix is 5. What are the possible values of the determinant of the matrix?

1) 0
2) 5
3) 10
4) Cannot be determined

Answer 1

The determinant of a 2x2 matrix with a trace of 5 cannot be determined because the determinant is influenced by all elements of the matrix, not just the trace.

Step-by-step explanation:

The trace of a 2x2 matrix is the sum of its diagonal elements, while the determinant is the product of its diagonal elements minus the product of its off-diagonal elements. For a 2x2 matrix A with elements a11, a12, a21, and a22, the trace is a11 + a22 and the determinant is a11a22 - a12a21. Since the trace is given as 5, the determinant can vary widely depending on the specific elements of the matrix, so the correct answer is that it cannot be determined with the information given.