Page 181 - 35Linear Algebra

8.2 Elementary Matrices and Determinants 181

Figure 8.5: Determinants measure if a matrix is invertible.

i
i
i
Corollary 8.2.3. Any elementary matrix E , R (λ), S (µ) is invertible, ex-
j
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cept for R (0). In fact, the inverse of an elementary matrix is another ele-
mentary 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
n
exists a row of zeros in RREF(M), so R (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)
n
= det(E 1 ) · · · det(E k ) det(R (λ) RREF(M)N)
= det(E 1 ) · · · det(E k )λ det(RREF(M)N)
= λ det(MN)

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