linear-guest

8.2 Elementary Matrices and Determinants 181
Figure 8.5: Determinants measure if a matrix is invertible.
Corollary 8.2.3. Any elementary matrix E i j, R i (λ), S i j(µ) is invertible, except
for R i (0). In fact, the inverse of an elementary matrix is another elementary
matrix.
To obtain one last important result, suppose that M and N are square
n × n matrices, with reduced row echelon forms such that, for elementary
matrices E i and F i ,
M = E 1 E 2 · · · E k RREF(M) ,
and
N = F 1 F 2 · · · F l RREF(N) .
If RREF(M) is the identity matrix (i.e., M is invertible), then:
det(MN) = det(E 1 E 2 · · · E k RREF(M)F 1 F 2 · · · F l RREF(N))
= det(E 1 E 2 · · · E k IF 1 F 2 · · · F l RREF(N))
= det(E 1 ) · · · det(E k ) det(I) det(F 1 ) · · · det(F l ) det RREF(N)
= det(M) det(N)
Otherwise, M is not invertible, and det M = 0 = det RREF(M). Then there
exists a row of zeros in RREF(M), so R n (λ) RREF(M) = RREF(M) for
any λ. Then:
det(MN) = det(E 1 E 2 · · · E k RREF(M)N)
= det(E 1 ) · · · det(E k ) det(RREF(M)N)
= det(E 1 ) · · · det(E k ) det(R n (λ) RREF(M)N)
= det(E 1 ) · · · det(E k )λ det(RREF(M)N)
= λ det(MN)
181