Example ¹3. Finding the Determinant of a 5x5 MatrixThis solution was made using the calculator presented on the site.
Let's calculate the determinant A using a elementary transformations.
| det A = |
| 4 | 1 | 1 | 2 | 1 | | = | | 1 | 2 | -1 | 1 | 1 | | 3 | 1 | 1 | 1 | 1 | | 2 | 1 | 1 | 4 | 1 | | 2 | -1 | 1 | 1 | 5 | The elements of row 3 multiplied by -1 are added to the corresponding elements of row 1. more info
| 4 + 3 * ( -1) | 1 + 1 * ( -1) | 1 + 1 * ( -1) | 2 + 1 * ( -1) | 1 + 1 * ( -1) | | | 1 | 2 | -1 | 1 | 1 | | 3 | 1 | 1 | 1 | 1 | | 2 | 1 | 1 | 4 | 1 | | 2 | -1 | 1 | 1 | 5 | This elementary transformation does not change the value of the determinant.
| = |
| 1 | 0 | 0 | 1 | 0 | | = | | 1 | 2 | -1 | 1 | 1 | | 3 | 1 | 1 | 1 | 1 | | 2 | 1 | 1 | 4 | 1 | | 2 | -1 | 1 | 1 | 5 | The elements of column 4 multiplied by -1 are added to the corresponding elements of column 1. more info
| 1 + 1 * ( -1) | 0 | 0 | 1 | 0 | | | 1 + 1 * ( -1) | 2 | -1 | 1 | 1 | | 3 + 1 * ( -1) | 1 | 1 | 1 | 1 | | 2 + 4 * ( -1) | 1 | 1 | 4 | 1 | | 2 + 1 * ( -1) | -1 | 1 | 1 | 5 | This elementary transformation does not change the value of the determinant.
| = |
| 0 | 0 | 0 | 1 | 0 | | = | | 0 | 2 | -1 | 1 | 1 | | 2 | 1 | 1 | 1 | 1 | | -2 | 1 | 1 | 4 | 1 | | 1 | -1 | 1 | 1 | 5 | Expand the determinant along the row 1. more info
| 0 | 0 | 0 | 1 | 0 | | | 0 | 2 | -1 | 1 | 1 | | 2 | 1 | 1 | 1 | 1 | | -2 | 1 | 1 | 4 | 1 | | 1 | -1 | 1 | 1 | 5 | |
Row number 1
Column number 1 |
|
Element |
|
Row 1 and column 1
have been deleted |
| ( -1) 1 + 1 |
* |
0 |
* |
| |
| 2 | -1 | 1 | 1 | | | | 1 | 1 | 1 | 1 | | 1 | 1 | 4 | 1 | | -1 | 1 | 1 | 5 | |
| 0 | 0 | 0 | 1 | 0 | | | 0 | 2 | -1 | 1 | 1 | | 2 | 1 | 1 | 1 | 1 | | -2 | 1 | 1 | 4 | 1 | | 1 | -1 | 1 | 1 | 5 | |
Row number 1
Column number 2 |
|
Element |
|
Row 1 and column 2
have been deleted |
| ( -1) 1 + 2 |
* |
0 |
* |
| |
| 0 | -1 | 1 | 1 | | | | 2 | 1 | 1 | 1 | | -2 | 1 | 4 | 1 | | 1 | 1 | 1 | 5 | |
| 0 | 0 | 0 | 1 | 0 | | | 0 | 2 | -1 | 1 | 1 | | 2 | 1 | 1 | 1 | 1 | | -2 | 1 | 1 | 4 | 1 | | 1 | -1 | 1 | 1 | 5 | |
Row number 1
Column number 3 |
|
Element |
|
Row 1 and column 3
have been deleted |
| ( -1) 1 + 3 |
* |
0 |
* |
| |
| 0 | 2 | 1 | 1 | | | | 2 | 1 | 1 | 1 | | -2 | 1 | 4 | 1 | | 1 | -1 | 1 | 5 | |
| 0 | 0 | 0 | 1 | 0 | | | 0 | 2 | -1 | 1 | 1 | | 2 | 1 | 1 | 1 | 1 | | -2 | 1 | 1 | 4 | 1 | | 1 | -1 | 1 | 1 | 5 | |
Row number 1
Column number 4 |
|
Element |
|
Row 1 and column 4
have been deleted |
| ( -1) 1 + 4 |
* |
1 |
* |
| |
| 0 | 2 | -1 | 1 | | | | 2 | 1 | 1 | 1 | | -2 | 1 | 1 | 1 | | 1 | -1 | 1 | 5 | |
| 0 | 0 | 0 | 1 | 0 | | | 0 | 2 | -1 | 1 | 1 | | 2 | 1 | 1 | 1 | 1 | | -2 | 1 | 1 | 4 | 1 | | 1 | -1 | 1 | 1 | 5 | |
Row number 1
Column number 5 |
|
Element |
|
Row 1 and column 5
have been deleted |
| ( -1) 1 + 5 |
* |
0 |
* |
| |
| 0 | 2 | -1 | 1 | | | | 2 | 1 | 1 | 1 | | -2 | 1 | 1 | 4 | | 1 | -1 | 1 | 1 | |
Products are summed. If the element is zero then product is zero too.
| = ( -1) 1 + 4 * 1 * |
| 0 | 2 | -1 | 1 | | = | | 2 | 1 | 1 | 1 | | -2 | 1 | 1 | 1 | | 1 | -1 | 1 | 5 |
| = - |
| 0 | 2 | -1 | 1 | | = | | 2 | 1 | 1 | 1 | | -2 | 1 | 1 | 1 | | 1 | -1 | 1 | 5 | The elements of row 2 multiplied by -1 are added to the corresponding elements of row 3. more info
| 0 | 2 | -1 | 1 | | | 2 | 1 | 1 | 1 | | -2 + 2 * ( -1) | 1 + 1 * ( -1) | 1 + 1 * ( -1) | 1 + 1 * ( -1) | | 1 | -1 | 1 | 5 | This elementary transformation does not change the value of the determinant.
| = - |
| 0 | 2 | -1 | 1 | | = | | 2 | 1 | 1 | 1 | | -4 | 0 | 0 | 0 | | 1 | -1 | 1 | 5 | Expand the determinant along the row 3. more info
| 0 | 2 | -1 | 1 | | | 2 | 1 | 1 | 1 | | -4 | 0 | 0 | 0 | | 1 | -1 | 1 | 5 | |
Row number 3
Column number 1 |
|
Element |
|
Row 3 and column 1
have been deleted |
| ( -1) 3 + 1 |
* |
-4 |
* |
|
| 0 | 2 | -1 | 1 | | | 2 | 1 | 1 | 1 | | -4 | 0 | 0 | 0 | | 1 | -1 | 1 | 5 | |
Row number 3
Column number 2 |
|
Element |
|
Row 3 and column 2
have been deleted |
| ( -1) 3 + 2 |
* |
0 |
* |
|
| 0 | 2 | -1 | 1 | | | 2 | 1 | 1 | 1 | | -4 | 0 | 0 | 0 | | 1 | -1 | 1 | 5 | |
Row number 3
Column number 3 |
|
Element |
|
Row 3 and column 3
have been deleted |
| ( -1) 3 + 3 |
* |
0 |
* |
|
| 0 | 2 | -1 | 1 | | | 2 | 1 | 1 | 1 | | -4 | 0 | 0 | 0 | | 1 | -1 | 1 | 5 | |
Row number 3
Column number 4 |
|
Element |
|
Row 3 and column 4
have been deleted |
| ( -1) 3 + 4 |
* |
0 |
* |
|
Products are summed. If the element is zero then product is zero too.
| = - ( ( -1) 3 + 1 * ( -4) * |
| 2 | -1 | 1 | | ) = | | 1 | 1 | 1 | | -1 | 1 | 5 |
| = 4 * |
| 2 | -1 | 1 | | = | | 1 | 1 | 1 | | -1 | 1 | 5 | The elements of row 2 multiplied by -1 are added to the corresponding elements of row 3. more info
| 2 | -1 | 1 | | | 1 | 1 | 1 | | -1 + 1 * ( -1) | 1 + 1 * ( -1) | 5 + 1 * ( -1) | This elementary transformation does not change the value of the determinant.
| = 4 * |
| 2 | -1 | 1 | | = | | 1 | 1 | 1 | | -2 | 0 | 4 | The elements of row 2 are added to the corresponding elements of row 1. more info
| 2 + 1 | -1 + 1 | 1 + 1 | | | 1 | 1 | 1 | | -2 | 0 | 4 | This elementary transformation does not change the value of the determinant.
| = 4 * |
| 3 | 0 | 2 | | = | | 1 | 1 | 1 | | -2 | 0 | 4 | Expand the determinant along the column 2. more info
|
Row number 1
Column number 2 |
|
Element |
|
Row 1 and column 2
have been deleted |
| ( -1) 1 + 2 |
* |
0 |
* |
|
|
Row number 2
Column number 2 |
|
Element |
|
Row 2 and column 2
have been deleted |
| ( -1) 2 + 2 |
* |
1 |
* |
|
|
Row number 3
Column number 2 |
|
Element |
|
Row 3 and column 2
have been deleted |
| ( -1) 3 + 2 |
* |
0 |
* |
|
Products are summed. If the element is zero then product is zero too.
| = 4 * ( -1) 2 + 2 * 1 * |
| 3 | 2 | | = | | -2 | 4 | = 4 * ( 3 * 4 - 2 * ( -2) ) = = 64
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