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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
...
Then the cofactors of the i-th column of N; will equal the cofactors of the i-th column
of A, and hence we get
det N;
=
b1Ci;+b2C2;+
· ·
·
+b11C11;
Therefore, the i-th component of 1 in the solution of Ax
x· -
=
b is
det N;
--
- detA
1
This is called Cramer's Rule (or Method)
...
EXAMPLE3
Use Cramer's Rule to solve the system of equations
...
s A
...
detA
Hence,
Xi
x
2
=
3
-X2
5
1
-x2
3
+
-
=
-
=
r-22/315
=
1
5
1
2
-
- · - - -·
-2 -1
3
3
3 2
5 5
1
1
1
1
_
3/5
11 3
=
-
]
, so
-4
225
-
165
-225
...
93 93
-2/3 1/5
-4/225 2/5 1/2
4
4
4
=
To solve a system of
n
equations in
n
_
=
=
variables by using Cramer's Rule would
require the evaluation of the determinant of
n +
1
matrices, each of which is
n x n
...
However, like the cofactor method above, Cramer's Rule is sometimes used
to write a formula for the solution of a problem
...
3
Practice Problems
Al Determine the inverse of each of the following ma
trices by using the cofactor method
...
(a)
(b)
(c)
[! l�]
[; =�1
U -� Il
(d)
[�
-2
A2 Let A
=
-
� �
2 -1
i
[-; -� �]·
-
3
1
1
(a) Determine the cofactor matrix cof A
...
(a) x
2 1-x
3 2=6
(c)
7x1+
x2 - x
4 3 =3
-x
6 1 - 4x2+
x3 =0
3x1 +x
5 2=7
4x1 -x2 - x
2 3 =6
(b) x
3 1+x
3 2=2
(d)
x
2 1-x
3 2=5
x
2 1+3x2 - 5x3 =2
3x1 -x2 +2x3 =1
5x1+4x2 - x
6 3 =3
Homework Problems
[ � � �1
...
Verify your
answer by using multiplication
...
(b) Calculate A(cof A)7 and determine
and A-1•
-2
2
-2
-
B3 Use Cramer's Rule to solve the following systems
...
(a) Verify by Cramer's Rule that the system of
equations Ax=a1 has the unique solution
x=ej (the }-th standard basis vector)
...
D2 Let A =
2
-1
0
0
-1
3
0
0
2
1
2
0
0
0
3
...
(If you calcu
late more than these two entries of A-1, you have
missed the point
Title: Matrix inverse by cofactors and Cramer's rule
Description: Linear algebra course
Description: Linear algebra course