b) By a), .
c) Use the hint. The subspace consisting of matrices in with trace 0 is the kernel of the map in the hint.
d) The identity shows that every n×n matrix can be written as the sum of a matrix with trace 0 (namely the expression in parentheses) and a multiple of the identity.
Write for the matrix having a 1 in the (i,j)-position and zeros everywhere else. If then . By a linear combination of such matrices you can construct any matrix with zeros along the diagonal as a linear combination of commutators. Next, for all . Check that by forming linear combinations of such matrices you can form any diagonal matrix that is not a scalar multiple of the identity.
That shows that any matrix with trace 0 can be expressed as sum of commutators.