|
| The
inverse of a matrix |
| The
inverse matrix A-1,
of a square matrix A,
is the matrix which when multiplied by A
on either side gives the
identity matrix I
(the
identity matrix is a square matrix in which every main diagonal
entry is 1 and the other
elements are all
zero), |
| AA-1 =
A-1A =
I . |
| The
inverse of a regular square matrix A = [aik]
(with a non zero determinant) we obtain dividing each entry of the transpose of the cofactor
matrix of A, called the adjoint matrix,
by the value of determinant of A,
i.e., |
 |
| The
entries of the cofactor matrix [Aik] |
| Aik =
(-1)i
+ k · Dik
, |
| where
Dik
is the minor (or subdeterminant) of the matrix A
obtained by deleting its ith
row and kth
column. |
| Interchange the rows and columns
of the cofactor matrix to
obtain [Aki]
the transpose of the cofactor matrix. |
|
| Example:
Find the inverse of the 3
´
3 matrix A
from the previous example.
|
| Solution:
By
expanding the det(A)
by the second row obtained is,
|
 |
| Then
calculate the cofactors Aik, |
 |
| to
write the cofactor matrix [Aik] |
 |
| By
flipping the cofactor matrix [Aik]
around the main diagonal
obtained is the adjoint
matrix [Aki]
or the transpose
of the cofactor matrix. |
| Therefore, the inverse of the
matrix A |
 |
| Let
verify that the matrix A
multiplied by its inverse gives the identity matrix, |
 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| Pre-calculus contents
K |
|
 |
|
|
|
Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved.
|
