4, Let M2(R) be the set of 2 × 2 matrices with real entries. That is: 021 a22 De

Question

4, Let M2(R) be the set of 2 × 2 matrices with real entries. That is: 021 a22 Define Tr : M2(R) R by Tr(A) = an +a22. This is called the trace of the matrix. (a) Find Tr5 (b) Find Tr (10,1)) (c) Is the trace function injective? Support your answers with proofs or counter examples (d) Is it surjective? Support your answers with proofs or counter examples. 5. Let M2(R) be as above. Let Det : M(R) R be defined by Det(A)-ana22-021 a 12. This is called the determinant of the matrix A. (a) Find 1 2 Det 3 6 (b) Express the condition to be in the set Det-(0) in terms of an,912,031,022 (c) Is the function Det injective? Support your answers with proofs or counter ex amples. (d) Is it surjective? Support your answers with proofs or counter examples

Explanation / Answer

4

a) Set of all matrices with trace =5

b) Set of all matrices with trace equal to 0 or 1

c) No.

eg.

[ 0 0]

[0 0]

and

[1 0]

[0 -1]

have trace equal to 0

d)

Yes.

We let all entries except a_11 to be 0 and vary a_11 over all R

5

a)

=1*6-3*3=0

b)

Matrix must be singular is non invertible

a_11a_22=a_12a_21

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