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Math 2331
Linear Algebra
Section 5.3
Cramer's Rule, Inverses and Volumes
Cramer's Rule
This is a way to solve Ax = b using determinants
If det(A) is not zero (i.e. A is square and invertible),
1. Ax = b is solvable for any b
2. Cramer's rule can solve for each component of x individually
To find the jth component of b, first replace the jth column of A with the vector b and
let's call this matrix
j
B
Then
det(
)
det( )
j
j
B
x
A
=
Example: Use Cramer's Rule to find the solution to
12
32
6
54
xx
−=
−+
=
8
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View Full Document Cramer's Rule also gives us a formula for the inverse of A:
Given an nxn matrix A
Let
be the vector that is the jth column of the nxn identity matrix (a 1 in the jth row)
j
e
Solve
j
A
xe
=
for all j between 1 and n.
The (i,j) entry in the inverse of A is the ith row of x
Why?
This is the cofactor
NOTE THE ORDER !!!
ji
C
Cramer's Rule for Inverses
1
det( )
ji
ij
C
A
A
−
=
Let
11
12
1
21
22
2
12
...
...
...
...
n
n
nn
n
n
CC
C
C
C
C
⎛⎞
⎜⎟
=
⎝⎠
Then
1
det( )
T
C
A
A
−
=
Once again  to find the i,j  entry in the matrix
1
A
−
Cross out row j and column i and find the determinant of the resulting matrix. Multiply
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This note was uploaded on 08/10/2011 for the course MATH 2331 taught by Professor Staff during the Spring '08 term at University of Houston.
 Spring '08
 Staff
 Linear Algebra, Algebra, Determinant

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