College Algebra, 2013a

8.5 Determinants and Cramer’s Rule 621
If we expand det (A 1 )alongthefirstrow,weget
det (A 1 ) =
([ −1 5
det
1 4
=
([ −1 5
det
1 4
]) ([ 0 5
− det
0 4
])
]) ([ 0 −1
+2det
0 1
Amazingly, this is none other than the C 11 cofactor of A. The reader is invited to check this, as
well as the claims that det (A 2 )=C 12 and det (A 3 )=C 13 . 7 (To see this, though it seems unnatural
to do so, expand along the first row.) Cramer’s Rule tells us
x 11 = det (A 1)
det(A) = C 11
det(A) , x 21 = det (A 2)
det(A) = C 12
det(A) , x 31 = det (A 3)
det(A) = C 13
det(A)
So the first column of the inverse matrix X is:
⎡
⎣
x 11
x 21
x 31
⎡
⎤
⎦ =
⎢
⎣
C 11
det(A)
C 12
det(A)
C 13
det(A)
⎤
=
⎥
⎦
⎡
1
⎣
det(A)
Notice the reversal of the subscripts going from the unknown to the corresponding cofactor of A.
This trend continues and we get
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
x 12
C
⎣ x 22
⎦
1
21
x 13
C
= ⎣ C 22
⎦ ⎣ x 23
⎦
1
31
= ⎣ C 32
⎦
det(A)
det(A)
x 32
C 23
x 33
C 33
Putting all of these together, we have obtained a new and surprising formula for A −1 , namely
A −1 =
⎡
1
⎣
det(A)
C 11
C 12
C 13
C 11 C 21 C 31
⎤
C 12 C 22 C 32
⎦
To see that this does indeed yield A −1 , we find all of the cofactors of A
⎤
⎦
])
And, as promised,
C 11 = −9, C 21 = −2, C 31 = 7
C 12 = 10, C 22 = 8, C 32 = −15
C 13 = 2, C 23 = −1, C 33 = −3
7 In a solid Linear Algebra course you will learn that the properties in Theorem 8.7 hold equally well if the word
‘row’ is replaced by the word ‘column’. We’re not going to get into column operations in this text, but they do make
some of what we’re trying to say easier to follow.