|
| The
determinant of a matrix |
| Determinants
are defined only for square matrices. |
| The
determinant D
of an n
´
n
matrix A,
also denoted det(A)
or |A|
is an ordered square array of elements |
 |
| the
value of which is given by an alternating sum of products of the
elements of A,
that can be obtained |
| by
using the method of expanding the determinant to
cofactors. |
|
| Properties of determinants |
| Use of the following properties simplify calculation of the value of
higher order determinants. |
| 1.
Interchanging (switching) two rows or columns changes the sign
of the determinant. |
| 2.
A determinant with a row or column of zeros has value 0. |
| 3.
A determinant with two rows or columns equal (or proportional) has value 0. |
| 4.
A common factor of all elements of a row or a column can be factored out. |
| 5.
The multiplication of a determinant by a scalar can be done by multiplying each element of any row or |
| column by the scalar. |
| 6.
The determinant of a triangular matrix is equal to the product
of the diagonal elements. |
| 7.
The value of a determinant will not change by adding multiples of
any column or row to any other column or |
| row. |
| 8.
A
determinant can be expanded along any row or column (without
affecting the determinant's value). |
|
| Therefore,
we use the above properties to set as many zeros in a row or column as possible to simplify |
| calculation of the value of
a determinant. |
|
| Example:
Let find the
value of the determinant of the matrix A
from the previous example applying the
|
| method of expanding the determinant to
cofactors and using properties of determinants. |
 |
| First,
added is the second row to the first writing the sum to the
first row. Then we add third row to second, |
| writing
the sum to the second row. Finally, the first column multiplied
by -1
we added to the second column |
| to
obtain the triangular matrix. |
| By
expanding the above triangular matrix by the first column
obtained is |
 |
| This
way proved is the 6th
property stating that the determinant of a triangular matrix is
equal to the product |
| of
its diagonal elements. |
|
| 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, |
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Matrices, determinants and systems of linear equations
contents
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