Let X be a matrix of n × m and X = (X1, X2), where X1 is n × k matrix and X2 is n × (m − k) matrix. Show that (a). The matrices X(X 0 X) −1X 0 and X1(X 0 1X1) −1X 0 1 are idempotent. (b). The matrix X(X 0 X) −1X 0 − X2(X 0 2X2) −1X 0 2 is idempotent. (c). Find the rank of the matrix X(X 0 X) −1X 0 − X2(X 0 2X2) −1X 0 2 .