A First Course in Linear Algebra, 2017a

118 Determinants
Solution. Consider the matrix A first. Using Definition 3.1 we can find the determinant as follows:
det(A)=3 × 4 − 2 × 6 = 12 − 12 = 0
By Theorem 3.28 A is not invertible.
Now consider the matrix B. Again by Definition 3.1 we have
det(B)=2 × 1 − 5 × 3 = 2 − 15 = −13
By Theorem 3.28 B is invertible and the determinant of the inverse is given by
det ( B −1) =
1
det(B)
=
1
−13
= − 1
13
♠
3.1.4 Properties of Determinants II: Some Important Proofs
This section includes some important proofs on determinants and cofactors.
First we recall the definition of a determinant. If A = [ ]
a ij is an n × n matrix, then detA is defined by
computing the expansion along the first row:
detA =
n
∑
i=1
a 1,i cof(A) 1,i . (3.1)
If n = 1thendetA = a 1,1 .
The following example is straightforward and strongly recommended as a means for getting used to
definitions.
Example 3.30:
(1) Let E ij be the elementary matrix obtained by interchanging ith and jth rows of I. ThendetE ij =
−1.
(2) Let E ik be the elementary matrix obtained by multiplying the ith row of I by k. ThendetE ik = k.
(3) Let E ijk be the elementary matrix obtained by multiplying ith row of I by k and adding it to its
jth row. Then detE ijk = 1.
(4) If C and B are such that CB is defined and the ith row of C consists of zeros, then the ith row of
CB consists of zeros.
(5) If E is an elementary matrix, then detE = detE T .
Many of the proofs in section use the Principle of Mathematical Induction. This concept is discussed
in Appendix A.2 and is reviewed here for convenience. First we check that the assertion is true for n = 2
(the case n = 1 is either completely trivial or meaningless).