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NOTES ON SECTION
3
.
4
AND
3
.
5
MA265, SECTIONS 41, 52
Key words
Inverse Adjoint (or adjugate) Cramer’s rule
1.
Inverse using determinants
‘ The point of 3
.
4 is a purely algebraic (but very complicated) expression for the inverse of a matrix. In
words, the inverse of a matrix
A
is the adjoint matrix (this terminology is unfortunate for reasons we will
not explain – wikipedia prefers the name “adjugate matrix”, as would I) scaled by 1
/
det
A
. The adjoint
matrix, in turn, is the
transpose
of the matrix of cofactors. That is
Deﬁnition 1.
The
adjoint
matrix of the matrix
A
is the matrix
adj
A
=
A
11
A
21
...
A
n
1
A
12
A
22
...
A
n
2
.
.
.
.
.
.
.
.
.
A
1
n
A
2
n
A
nn
where the
A
ij
are the cofactors of the entries
a
ij
of the matrix
A
.
Remark 2.
Note carefully the strange ordering of the entries!!! It is the transpose of what you would write
if you simply replaced the entries of the matrix with its cofactor.
As I was saying,
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This note was uploaded on 03/31/2012 for the course MA 265 taught by Professor Bens during the Spring '08 term at Purdue UniversityWest Lafayette.
 Spring '08
 Bens
 Linear Algebra, Algebra, Determinant

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