Can you give me further explanation on these proofs

? Exercise 3.E. 14 Let A be an r x r matrix and suppose there are r - 1 rows (columns) such that all rows (columns) are linear combinations of these r - 1 rows (columns). Show det(A) = 0. Answer If the determinant is nonzero, then it will remain nonzero with row operations applied to the matrix. However, by assumption, you can obtain a row of zeros by doing row operations. Thus the determinant must have been zero after all. ? Exercise 3.E. 15 Show det(aA) = a" det(A) for an n x n matrix A and scalar a. Answer det(aA) = det(aIA) = det(al) det(A) = a" det (A) . The matrix which has a down the main diagonal has determinant equal to a"