Page 469 - Matrix Analysis & Applied Linear Algebra

6.1 Determinants 465

In particular, (6.1.9)–(6.1.11) guarantee that the determinants of elementary
matrices of Types I, II, and III are nonzero.
As discussed in §3.9, if P is an elementary matrix of Type I, II, or III,
and if A is any other matrix, then the product PA is the matrix obtained by
performing the elementary operation associated with P to the rows of A. This,
together with the observations (6.1.5)–(6.1.7) and (6.1.9)–(6.1.11), leads to the
conclusion that for every square matrix A,

det (EA)= −det (A) = det (E)det (A),
det (FA) = α det (A) = det (F)det (A),
det (GA) = det (A) = det (G)det (A).

In other words, det (PA) = det (P)det (A) whenever P is an elementary matrix
of Type I, II, or III. It’s easy to extend this observation to any number of these
elementary matrices, P 1 , P 2 ,..., P k , by writing

det (P 1 P 2 ··· P k A) = det (P 1 )det (P 2 ··· P k A)
= det (P 1 )det (P 2 )det (P 3 ··· P k A)
. (6.1.12)
.
.
= det (P 1 )det (P 2 ) ··· det (P k )det (A).
This leads to a characterization of invertibility in terms of determinants.

Invertibility and Determinants

• A n×n is nonsingular if and only if det (A) = 0 (6.1.13)
or, equivalently,
• A n×n is singular if and only if det (A)=0. (6.1.14)

Proof. Let P 1 , P 2 ,..., P k be a sequence of elementary matrices of Type I, II,
or III such that P 1 P 2 ··· P k A = E A , and apply (6.1.12) to conclude

det (P 1 )det (P 2 ) ··· det (P k )det (A) = det (E A ).

Since elementary matrices have nonzero determinants,

det (A) =0 ⇐⇒ det (E A ) =0 ⇐⇒ there are no zero pivots
⇐⇒ every column in E A (and in A) is basic
⇐⇒ A is nonsingular.