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Title: Matrix inverse by cofactors and Cramer's rule
Description: Linear algebra course
Description: Linear algebra course
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Matrix Inverse by Cofactors
and Cramer's Rule
5
...
Determinant calculations are generally much longer than
row reduction
...
However, it is useful in some theoretical applications
because it provides a formula for A-1 in terms of the entries of A
...
[False Expansion Theorem]
Theorem l
If A is an n
x
n matrix and i t= k, then
Proof: Let B be the matrix obtained from A by replacing (not swapping) the k-th row
of A by the i-th row of A
...
2
...
Since the cofactors c;j of B are equal to the cofactors
CkJ of A, and the coefficients bkJ of the k-th row of B are equal to the coefficients aiJ
of the i-th row of A, we get
as required
...
We define the cofactor matrix of A, denoted cof A, by
(cof A)iJ
=
CiJ
EXAMPLE 1
Let A=
2[0 4 -1 1
-2
1
...
5
3
6
Solution: The nine cofactors of A are
=Cl)1-� �I 17 12 I� �I=
1
1
2
C21 14_2 1= -18 C22= 1 1=
-11=7 C32=(-1)012 -11 =-2
=
ell
= (-1)
5
c
= c-1)
( 1)
C13=(1)I� _;1= -18
C23= I� -�I= 28
C33 (1)1� �I
6
5
6
1
(-1)
16
=
l
=6
Hence,
= [-1
EXERCISE 1
Calculate the cofactor matrix of A
�n
Observe that the cofactors of the i-th row of A form the i-th row of cofA, so they
form the i-th column of (cof Al
...
Moreover, by the False Expansion
Theorem, the dot product of the i-th row of A and the }-th column of cof A equals
if i * j
...
/
[c1
c,!]
-+T
a
i11 ·en
n
=
a11
•
C1
detA
0
=
0
where I is the identity matrix
...
en
detA
(detA)/
detA
(de;A) (cofAl= I, and, therefore,
A-1
=( � )
de A
ccofA)7
If detA
=
0, then, by Theorem 5
...
4, A is not invertible
...
(Some people refer to the trans
pose of the cofactor matrix as the adjugate matrix and therefore call this the adjugate
method)
...
i
2
4
-1
0
3
1
0
-14
8
2(24 + 14)
=
=
76
Thus, A is invertible
...
This is manageable, but it is more work than would be required by the
row reduction method
...
Cramer's Rule
Consider the system of
n
linear equations in
n
variables, Ax
=
that A is invertible, then the solution may be written in the form
x
=
A
-
1
b
X1
X;
Xn
=
detA
1
=
detA
(cof A)
7
b
C11
C 21
C111
b1
C1;
C2;
c,li
b;
C111
C211
Cnn
bn
b
...
+bn C)
Ill
I
I
detA
·
multiplied by the dot product of the vector
b with the i-th row of (cof Af
...
So
x; is the dot product of the vector
b with the i-th column of cof A divided by detA
Title: Matrix inverse by cofactors and Cramer's rule
Description: Linear algebra course
Description: Linear algebra course