Page 180 - 35Linear Algebra

180 Determinants

We have seen that any matrix M can be put into reduced row echelon form
via a sequence of row operations, and we have seen that any row operation can
be achieved via left matrix multiplication by an elementary matrix. Suppose
that RREF(M) is the reduced row echelon form of M. Then

RREF(M) = E 1 E 2 · · · E k M ,

where each E i is an elementary matrix. We know how to compute determi-
nants of elementary matrices and products thereof, so we ask:

What is the determinant of a square matrix in reduced row echelon form?

The answer has two cases:

1. If M is not invertible, then some row of RREF(M) contains only zeros.
Then we can multiply the zero row by any constant λ without chang-
ing M; by our previous observation, this scales the determinant of M
by λ. Thus, if M is not invertible, det RREF(M) = λ det RREF(M),
and so det RREF(M) = 0.

2. Otherwise, every row of RREF(M) has a pivot on the diagonal; since
M is square, this means that RREF(M) is the identity matrix. So if
M is invertible, det RREF(M) = 1.

Notice that because det RREF(M) = det(E 1 E 2 · · · E k M), by the theorem
above,
det RREF(M) = det(E 1 ) · · · det(E k ) det M .

Since each E i has non-zero determinant, then det RREF(M) = 0 if and only
if det M = 0. This establishes an important theorem:

Theorem 8.2.2. For any square matrix M, det M 6= 0 if and only if M is
invertible.

Since we know the determinants of the elementary matrices, we can im-
mediately obtain the following:

Determinants and Inverses

180