Example of Finding the Determinant of a 4x4 Matrix
This solution was made using the calculator presented on the site.
Let's calculate the determinant A using a elementary transformations.
| det A = |
| 3 | -3 | -5 | 8 | | = |
| -3 | 2 | 4 | -6 |
| 2 | -5 | -7 | 5 |
| -4 | 3 | 5 | -6 |
The elements of row 2 multiplied by -1 are added to the corresponding elements of row 4.
more info
| 3 | -3 | -5 | 8 | |
| -3 | 2 | 4 | -6 |
| 2 | -5 | -7 | 5 |
| -4 + ( -3) * ( -1) | 3 + 2 * ( -1) | 5 + 4 * ( -1) | -6 + ( -6) * ( -1) |
This elementary transformation does not change the value of the determinant.
| = |
| 3 | -3 | -5 | 8 | | = |
| -3 | 2 | 4 | -6 |
| 2 | -5 | -7 | 5 |
| -1 | 1 | 1 | 0 |
The elements of column 1 are added to the corresponding elements of column 2.
more info
| 3 | -3 + 3 | -5 | 8 | |
| -3 | 2 + ( -3) | 4 | -6 |
| 2 | -5 + 2 | -7 | 5 |
| -1 | 1 + ( -1) | 1 | 0 |
This elementary transformation does not change the value of the determinant.
| = |
| 3 | 0 | -5 | 8 | | = |
| -3 | -1 | 4 | -6 |
| 2 | -3 | -7 | 5 |
| -1 | 0 | 1 | 0 |
The elements of column 1 are added to the corresponding elements of column 3.
more info
| 3 | 0 | -5 + 3 | 8 | |
| -3 | -1 | 4 + ( -3) | -6 |
| 2 | -3 | -7 + 2 | 5 |
| -1 | 0 | 1 + ( -1) | 0 |
This elementary transformation does not change the value of the determinant.
| = |
| 3 | 0 | -2 | 8 | | = |
| -3 | -1 | 1 | -6 |
| 2 | -3 | -5 | 5 |
| -1 | 0 | 0 | 0 |
Expand the determinant along the row 4.
more info
| 3 | 0 | -2 | 8 | | | -3 | -1 | 1 | -6 | | 2 | -3 | -5 | 5 | | -1 | 0 | 0 | 0 | |
Row number 4
Column number 1 |
|
Element |
|
Row 4 and column 1
have been deleted |
| ( -1) 4 + 1 |
* |
-1 |
* |
|
| 3 | 0 | -2 | 8 | | | -3 | -1 | 1 | -6 | | 2 | -3 | -5 | 5 | | -1 | 0 | 0 | 0 | |
Row number 4
Column number 2 |
|
Element |
|
Row 4 and column 2
have been deleted |
| ( -1) 4 + 2 |
* |
0 |
* |
|
| 3 | 0 | -2 | 8 | | | -3 | -1 | 1 | -6 | | 2 | -3 | -5 | 5 | | -1 | 0 | 0 | 0 | |
Row number 4
Column number 3 |
|
Element |
|
Row 4 and column 3
have been deleted |
| ( -1) 4 + 3 |
* |
0 |
* |
|
| 3 | 0 | -2 | 8 | | | -3 | -1 | 1 | -6 | | 2 | -3 | -5 | 5 | | -1 | 0 | 0 | 0 | |
Row number 4
Column number 4 |
|
Element |
|
Row 4 and column 4
have been deleted |
| ( -1) 4 + 4 |
* |
0 |
* |
|
Products are summed. If the element is zero than product is zero too.
| = ( -1) 4 + 1 * ( -1) * |
| 0 | -2 | 8 | | = |
| -1 | 1 | -6 |
| -3 | -5 | 5 |
| = |
| 0 | -2 | 8 | | = |
| -1 | 1 | -6 |
| -3 | -5 | 5 |
The elements of row 2 multiplied by -3 are added to the corresponding elements of row 3.
more info
| 0 | -2 | 8 | |
| -1 | 1 | -6 |
| -3 + ( -1) * ( -3) | -5 + 1 * ( -3) | 5 + ( -6) * ( -3) |
This elementary transformation does not change the value of the determinant.
| = |
| 0 | -2 | 8 | | = |
| -1 | 1 | -6 |
| 0 | -8 | 23 |
Expand the determinant along the column 1.
more info
|
Row number 1
Column number 1 |
|
Element |
|
Row 1 and column 1
have been deleted |
| ( -1) 1 + 1 |
* |
0 |
* |
|
|
Row number 2
Column number 1 |
|
Element |
|
Row 2 and column 1
have been deleted |
| ( -1) 2 + 1 |
* |
-1 |
* |
|
|
Row number 3
Column number 1 |
|
Element |
|
Row 3 and column 1
have been deleted |
| ( -1) 3 + 1 |
* |
0 |
* |
|
Products are summed. If the element is zero than product is zero too.
| = ( -1) 2 + 1 * ( -1) * |
| -2 | 8 | | = |
| -8 | 23 |
= 18
