# Service for Solving Linear Programming Problems

and other interesting typical problems
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## Example №2. Finding the Determinant of a 4x4 Matrix

This solution has been made using the calculator presented on the site.
Let's calculate the determinant A using a elementary transformations.
 det A = 4 6 -2 4 = 1 2 -3 1 4 -2 1 0 6 4 4 6
The elements of column 1 multiplied by -1 are added to the corresponding elements of column 4.   more info
 4 6 -2 4 + 4 * ( -1) 1 2 -3 1 + 1 * ( -1) 4 -2 1 0 + 4 * ( -1) 6 4 4 6 + 6 * ( -1)
This elementary transformation does not change the value of the determinant.
 = 4 6 -2 0 = 1 2 -3 0 4 -2 1 -4 6 4 4 0
 4 6 -2 0 1 2 -3 0 4 -2 1 -4 6 4 4 0
Row number 1
Column number 4
Element Row 1 and column 4
have been deleted
( -1) 1 + 4 * 0 *
 1 2 -3 4 -2 1 6 4 4
 4 6 -2 0 1 2 -3 0 4 -2 1 -4 6 4 4 0
Row number 2
Column number 4
Element Row 2 and column 4
have been deleted
( -1) 2 + 4 * 0 *
 4 6 -2 4 -2 1 6 4 4
 4 6 -2 0 1 2 -3 0 4 -2 1 -4 6 4 4 0
Row number 3
Column number 4
Element Row 3 and column 4
have been deleted
( -1) 3 + 4 * -4 *
 4 6 -2 1 2 -3 6 4 4
 4 6 -2 0 1 2 -3 0 4 -2 1 -4 6 4 4 0
Row number 4
Column number 4
Element Row 4 and column 4
have been deleted
( -1) 4 + 4 * 0 *
 4 6 -2 1 2 -3 4 -2 1
Products are summed. If the element is zero then product is zero too.
 = ( -1) 3 + 4 * ( -4) * 4 6 -2 = 1 2 -3 6 4 4
 = 4 * 4 6 -2 = 1 2 -3 6 4 4
The elements of column 1 multiplied by -1 are added to the corresponding elements of column 2.   more info
 4 6 + 4 * ( -1) -2 1 2 + 1 * ( -1) -3 6 4 + 6 * ( -1) 4
This elementary transformation does not change the value of the determinant.
 = 4 * 4 2 -2 = 1 1 -3 6 -2 4
The elements of column 2 multiplied by -2 are added to the corresponding elements of column 1.   more info
 4 + 2 * ( -2) 2 -2 1 + 1 * ( -2) 1 -3 6 + ( -2) * ( -2) -2 4
This elementary transformation does not change the value of the determinant.
 = 4 * 0 2 -2 = -1 1 -3 10 -2 4
 0 2 -2 + 2 -1 1 -3 + 1 10 -2 4 + ( -2)
This elementary transformation does not change the value of the determinant.
 = 4 * 0 2 0 = -1 1 -2 10 -2 2
 0 2 0 -1 1 -2 10 -2 2
Row number 1
Column number 1
Element Row 1 and column 1
have been deleted
( -1) 1 + 1 * 0 *
 1 -2 -2 2
 0 2 0 -1 1 -2 10 -2 2
Row number 1
Column number 2
Element Row 1 and column 2
have been deleted
( -1) 1 + 2 * 2 *
 -1 -2 10 2
 0 2 0 -1 1 -2 10 -2 2
Row number 1
Column number 3
Element Row 1 and column 3
have been deleted
( -1) 1 + 3 * 0 *
 -1 1 10 -2
Products are summed. If the element is zero then product is zero too.
 = 4 * ( -1) 1 + 2 * 2 * -1 -2 = 10 2
 = - 8 * -1 -2 = 10 2
= - 8 * ( -1 * 2 - ( -2) * 10 ) =
= - 8 * ( -2 + 20 ) =
= -144