5.4. THE DETERMINANT 73

5.4.6 The Determinant Of A ProductRecall the following definition of matrix multiplication.

Definition 5.4.9 If A and B are n×n matrices, A = (ai j) and B = (bi j), AB = (ci j) where

ci j ≡n

∑k=1

aikbk j.

One of the most important rules about determinants is that the determinant of a productequals the product of the determinants.

Theorem 5.4.10 Let A and B be n×n matrices. Then

det(AB) = det(A)det(B) .

Proof: Let ci j be the i jth entry of AB. Then by Proposition 5.4.3,

det(AB) =

∑(k1,··· ,kn)

sgn(k1, · · · ,kn)c1k1 · · ·cnkn

= ∑(k1,··· ,kn)

sgn(k1, · · · ,kn)

(∑r1

a1r1 br1k1

)· · ·

(∑rn

anrnbrnkn

)= ∑

(r1··· ,rn)∑

(k1,··· ,kn)

sgn(k1, · · · ,kn)br1k1 · · ·brnkn (a1r1 · · ·anrn)

= ∑(r1··· ,rn)

sgn(r1 · · ·rn)a1r1 · · ·anrn det(B) = det(A)det(B) .

5.4.7 Cofactor ExpansionsLemma 5.4.11 Suppose a matrix is of the form

M =

(A ∗0 a

)(5.4.13)

or

M =

(A 0∗ a

)(5.4.14)

where a is a number and A is an (n−1)× (n−1) matrix and ∗ denotes either a columnor a row having length n− 1 and the 0 denotes either a column or a row of length n− 1consisting entirely of zeros. Then det(M) = adet(A) .

Proof: Denote M by (mi j) . Thus in the first case, mnn = a and mni = 0 if i ̸= n while inthe second case, mnn = a and min = 0 if i ̸= n. From the definition of the determinant,

det(M)≡ ∑(k1,··· ,kn)

sgnn (k1, · · · ,kn)m1k1 · · ·mnkn