The problem is: \"Let M2(R) be the set of 2 X 2 matrices with real entries. That

Question

The problem is:

"Let M2(R) be the set of 2 X 2 matrices with real entries. That is, M2(R) = { A = [a11 a12][a21 a22] | a11,a12,a21,a22 are in R} (Note: R is the set of real numbers, row 1 is [a11  a12] , row 2 is [a21 a22] )

Let Det: M2(R) -->R be defined by Det(A) = a11a22 - a21a12. (This is the determinant of A).

(a) Is the function Det injective?

(b) Is the function Det surjective?

Support answers with proofs or counterexamples."

Please explain your answers. I'd appreciate anyone's help, and I'll be sure to give the points to the best answer. Thanks!

Explanation / Answer

(a) Det is not injective, for note that

A = [1 0][0 1]

B = [2 0][0 0.5]

A is not equal to B but

det (A) = (1)(1) - (0)(0) = 1

det(B) = (2)(.5) - (0)(0) = 1 = det(A)

Since two unequal matrices have the same determinant, Det is not injective.

(b) Det is surjective. Let r be a real number. We will find a matrix whose determinant is r.

Let A = [r 0][0 1]

Then det(A) = (r)(1) - (0)(0) = r - 0 = r

Thus, every real number r has a matrix whose determinant is r. Thus Det is surjective.

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