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Matrix Inversion
Definition
The inverse of an n by n matrix A is an n by n matrix B where,
AB = BA = In.
Please note: Not all matrices have inverses! Singular matrices don’t have inverse.
If a matrix has an inverse, then it is called invertible.
Definition
If A is a square matrix, and if a matrix B of the same size can be found such that AB=BA=I , then A is said to be invertible and B is called an inverse of A . If no such matrix B can be found, then A is said to be singular .
Notation:
1AB
Properties of Matrix Inverse
If A is an invertible matrix then its inverse is unique. (A-1)-1 = A. (Ak)-1= (A-1)k (we will denote this as A-k ) (cA)-1 = (1/c)A-1, c ≠ 0. ( AT)-1 = (A-1)T. If A is an invertible matrix, then the system of
equations Ax = b has a unique solution given by x = A-1b.
The Transpose of a Matrix A is the m×n matrix given by
The transpose matrix of A, denoted by AT, is a n×m matrix given by
.
21
22221
11211
mnmm
n
n
aaa
aaa
aaa
A
,
21
22212
12111
mnnn
m
m
T
aaa
aaa
aaa
A
The inverse of a 2-by-2 matrix
For a 2-by-2 matrix,
dc
baA
ac
bd
bcadA
11
The inverse of a 2 x 2 matrix
From this we deduced that a 2x2 matrix A is singular if and only if ad-bc = 0.
This quantity (ad-bc) has some other useful properties as well and so is defined to be the determinant of the matrix A.
Determinants of larger matrices
There is no “nice” formula for the inverse of larger than 2x2 matrices.
We still can define the determinant of a larger square matrix and it will still have the property that the determinant of A= 0 if and only if A is singular.
First we need some terminology.
Minors and cofactors
If A is a square matrix, then the minor Mij of the element aij of A is the determinant of the matrix obtained by deleting the i-th row and the j-th column from A.
The cofactor Cij = (-1)i+jMij.
Definition of a Determinant
If A is a square matrix of order 2 or greater, then the determinant of A is the sum of the entries in the first row of A multiplied by their cofactors. That is,
n
jjjCaAA
111)det(
Determinant of 2-by-2 Matrix
dc
baA
bcadAA )det(
Determinant of 3-by-3 Matrix
ihg
fed
cba
A
)()(
)det(
idbhfageccdhbfgaei
cegbfgcdhbdiafhaei
hg
edc
ig
fdb
ih
feaAA
Matrix Inversion
How to calculate the matrix inverse?
What is an adjoint matrix?
)(11 AadjA
A
Adjoint Matrix – Minors and Cofactors
The adjoint matrix of [A], Adj[A] is obtained by taking the transpose of the cofactor matrix of [A].
The minor for element aij of matrix [A] is found by removing the ith row and jth column from [A] and then calculating the determinant of the remaining matrix.
Matrix Inversion
1 2 3
1 2 3
1 2 3
2 4 5 36
-3 5 7 7
5 3 8 -31
x x x
x x x
x x x
Consider the following set of linear equations.
The coefficients can be arranged in a matrix form as shown.
2 -4 5
-3 5 7
5 3 -8
A
Matrix Inversion
1
2
3
2 -4 5 36
-3 5 7 7
5 3 -8 -31
x
x
x
[ ][ ] [ ]A x B=
The set of equations in matrix form is:
[ ] [ ] [ ]1x A B-=
Minors
835
753
542
835
753
542
835
753
542
11
5 7-61
3 -8M = =
12
-3 7-11
5 -8M = =
13
-3 5-34
5 3M = =
Minors
-61 -11 -34
17 -41 26
-53 29 -2
M
23
2 -426
5 3M
The resulting matrix of minors is:
835
753
542
Cofactors
Cofactors are the signed minors. The cofactor of element aij of matrix [A] is:
Therefore
The resulting matrix of cofactors is:
( )1 3
13 13-1C M+=
( )-1 i j
ij ijC M+=
( )1 2
12 12-1C M+=
( )1 1
11 11-1C M+=
-61 11 -34
-17 -41 -26
-53 -29 -2
C
Adjoint matrixThe adjoint matrix of [A], Adj[A] is obtained by taking the transpose of the cofactor matrix of [A].
-61 11 -34
-17 -41 -26
-53 -29 -2
C
Evaluate the determinant
-61 -17 -53
11 -41 -29
-34 -26 -2
adj A
2 -4 5
-3 5 7 -336
5 3 -8
A = =
[ ] [ ]Tadj A C=
Matrix Inversion [ ] [ ]1 1A adj A
A- =
1
-61 -17 -53
-336 -336 -33611 -41 -29
-336 -336 -336-34 -26 -2
-336 -336 -336
A
1
-61 -17 -531
11 -41 -29-336
-34 -26 -2
A
1
61 17 53
336 336 336-11 41 29
336 336 33617 13 1
168 168 168
A
Matrix Inversion Using G-J Elimination
If Gauss–Jordan elimination is applied on a square matrix, it can be used to calculate the matrix's inverse. This can be done by augmenting the square matrix with the identity matrix of the same dimensions., and through the following matrix operations:
IA IAA 1 1AI
210
121
012
A
If the original square matrix, A, is given by the following expression:
Then, after augmenting by the identity, the following is obtained:
100
010
001
210
121
012
IA
4
3
2
1
4
12
11
2
14
1
2
1
4
3
100
010
0011AI
100
010
001
I
By performing elementary row operations on the [AI] matrix until A reaches reduced row echelon form, the following is the final result:
The matrix augmentation can now be undone, which gives the following:
4
3
2
1
4
12
11
2
14
1
2
1
4
3
1A