If A is a square matrix of order n and m ϵ R then the det(mA) = (mn) (detA)
Well, observe that for any scalar and nxn matrix A.
where I is the standard nxn identity matrix. So you got . So you know the trick for determinats when multiplying matricies, Det(AB) = Det(A)Det(B). Well, the determinant of any matrix with entries only in the diagonals and zeroes everywhere is just the whole diagonal multiplied, since there are n diagonal enteries (all of them m),
so you got
Alternatively, it is easy to see that, for a 2 by 2 matrix, has determinant ad- bc. Multiplying by m gives matrix which has determinant . Now use the idea of expanding a determinant by columns to do a proof by induction on the size of the matrix.
Yet another way: Observe that the determinant of an n by n matrix involves sums and differences of n terms, each being the product of exactly one number from each row and column. With n rows and columns, each term will be a product of n numbers so each term in the determinant of m times the matrix will have a product of n numbers and so will have a factor of .
but this is a diagonal matrix, which has determinant: (1)(1)...(m)....(1) = m.
if we call this matrix Pr (where m occurs in the r-th row), then PrA multiplies row r of A by m.
hence mA = P1P2...PnA, so that:
det(mA) = det(P1)det(P2)...det(Pn)det(A) = (m)(m)....(m)(det(A)) = mn(det(A))
Here's a link:
Determinant with Row Multiplied by Constant - ProofWiki
This page shows all but one step of the proof you want, in formal language.
In general, proofwiki is a good place to look for proofs.
BTW, the idea of a determinant as a cofactor expansion was mysterious to me for a long time. I could do the expansions, but I didn't feel I understood why it worked, though I was shy about saying so. I've found that many other people are in the same boat. Suggestion: Google "Geometric Algebra Primer" by Jaap Suter, and/or watch the first few eps of Norman Wildberger's "Wild Lin Alg," which is on Youtube.
geometrically, what is happening is this:
multiplying ONE row of A by m is the same as stretching one SIDE of the n-dimensional volume element by a factor of m (which magnifies the volume element by a factor of m). if we do this for every side, we've stretched by a factor of mn
(it's easiest to comprehend this when n = 2 or 3).
There is another way to do it: Schur decomposition. Schur's theorem says that any square matrix A is unitarily equivalent to an upper-triangular matrix--i.e., A = U*TU, where U is unitary and T is upper-triangular. A and T have the same determinant (it's a fairly easy proof to show this, if needed), and since the determinant of a triangular matrix (upper or lower) equals the product of the main diagonal, multiplying each entry by m will multiply the overall determinant by m^n.