Unformatted text preview: st row.) Cramer’s Rule tells us
x11 = det (A1 )
C11
det (A2 )
C12
det (A3 )
C13
=
, x21 =
=
, x31 =
=
det(A)
det(A)
det(A)
det(A)
det(A)
det(A) So the ﬁrst column of the inverse matrix X is:
C11 det(A) C C
1 11 12 =
C12
= det(A) det(A)
C13 C13 det(A) x11 x21
x31 Notice the reversal of the subscripts going from the unknown to the corresponding cofactor of A.
This trend continues and we get C
x13
C
x12
1 31 1 21 x23 = x22 =
C22
C32
det(A)
det(A)
C23
x33
C33
x32
Putting all of these together, we have obtained a new C
1 11
C12
A−1 =
det(A)
C13 and surprising formula for A−1 , namely C21 C31
C22 C32 C23 C33 To see that this does indeed yield A−1 , we ﬁnd all of the cofactors of A
C11 = −9, C21 = −2, C31 =
7
C12 = 10, C22 =
8, C32 = −15
C13 =
2, C23 = −1, C33 = −3
And, as promised,
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...
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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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