Set up a system of three equations with two unknowns (alpha and beta). Is there a solution to these equations?
You need to check that
(a) T(x+y) = T(x) + T(y) when x and y are vectors in R^2; and
(b) T(cx) = cT(x) when c is a scalar and x is a vector in R^2.
Put the coefficient matrix into reduced row echelon form.
Multiply out AB and BA in terms of a and b. By setting the corresponding elements to be equal to one another, you have four linear equations in four unknowns. Find a solution.
For part (b), first do the multiplication without substituting for I and J. You can then replace J^2 with -I and simplify. Finally, you can substitute in for I and J and simplify.
You can write out matrices A = [a b; b c] and B = [d e;e f ]. Carry out the multiplication. The off diagaonal elements should not be equal. There are many choices.
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