Consider the following problem.

Let M denote the set of 2x2 real matrices. Let A be an element of M with trace 2 and determinant -3. Identifying M with R^{4}, consider the linear transformation T: M -> M defined by T(B) = AB. Then which of the following statements are true?

a) T is diagonalizable. b) 2 is an eigenvalue of T c) T is invertible d) T(B) = B for some non-zero matrix B in the set M.

Based on the opinion that thematrixof a linear transformation is thematrixwhich is multiplied by theinput matrixof the domain, to get theoutput matrixbelonging to the co-domain.

Here the definition of the transformation gives us the impression that it is obvious that the matrix A is the matrix of the linear transformation. So as per the given information since determinant of A is not zero it is invertible, which means it is also diagonalizable. Therefore options a and c are true. Since the trace is 2 and the determinant is -3, the two eigen values are -3 and +1 which shows that option b is not true. Similarly, since A cannot be the identity matrix, T(B) = AB can never be B itself. This means option d is also not true.

My question is this.

What does the phrase,"Identifying M with Rhave to do with the problem?^{4}"

Is there something overlooked by me due to my ignoring of the significance of this phrase?