| 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 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. |
|
|
|
Method of expanding
a determinant
of a rank n
by
cofactors, example
|
|
The value of a determinant will not change by adding multiples of
any column or row to any other column or row. This way created are zero entries that simplify subsequent calculations.
|
|
|
|
Example:
An application of the method of expanding a determinant to
cofactors to evaluate the determinant of the rank four.
|
|
|
Added is third to the second colon. Then, the second row multiplied
by -3 is added to the first row. The obtained determinant is then
expanded to its cofactors along the second colon:
|
|
|
The first colon multiplied by
-1 is added to the third colon. The
obtained determinant is then expanded along the third colon.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| Pre-calculus contents
K |
|
 |
|
|
|
Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved.
|
