Service for Solving Linear Programming Problems

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Example 1. Finding the Determinant of a 3x3 Matrix

This solution has been made using the calculator presented on the site.
1. Let's calculate the determinant A using a elementary transformations.
det A = -1 3 1 =
5 5 3
4 6 5
The elements of row 1 multiplied by -2 are added to the corresponding elements of row 3.   more info
-1 3 1
5 5 3
4 + ( -1) * ( -2) 6 + 3 * ( -2) 5 + 1 * ( -2)
This elementary transformation does not change the value of the determinant.
= -1 3 1 =
5 5 3
6 0 3
The elements of column 3 multiplied by -2 are added to the corresponding elements of column 1.   more info
-1 + 1 * ( -2) 3 1
5 + 3 * ( -2) 5 3
6 + 3 * ( -2) 0 3
This elementary transformation does not change the value of the determinant.
= -3 3 1 =
-1 5 3
0 0 3
Expand the determinant along the row 3.   more info
-3 3 1
-1 5 3
0 0 3
Row number 3
Column number 1
Element Row 3 and column 1
have been deleted
( -1) 3 + 1 * 0 *
3 1
5 3
-3 3 1
-1 5 3
0 0 3
Row number 3
Column number 2
Element Row 3 and column 2
have been deleted
( -1) 3 + 2 * 0 *
-3 1
-1 3
-3 3 1
-1 5 3
0 0 3
Row number 3
Column number 3
Element Row 3 and column 3
have been deleted
( -1) 3 + 3 * 3 *
-3 3
-1 5
Products are summed. If the element is zero then product is zero too.
= ( -1) 3 + 3 * 3 * -3 3 =
-1 5
= 3 * -3 3 =
-1 5
= 3 * ( -3 * 5 - 3 * ( -1) ) =
= 3 * ( -15 + 3 ) =
= -36
2. Let's calculate the determinant A by expanding along the row 1.
det A = -1 3 1 =
5 5 3
4 6 5
Expand the determinant along the row 1.   more info
-1 3 1
5 5 3
4 6 5
Row number 1
Column number 1
Element Row 1 and column 1
have been deleted
( -1) 1 + 1 * -1 *
5 3
6 5
-1 3 1
5 5 3
4 6 5
Row number 1
Column number 2
Element Row 1 and column 2
have been deleted
( -1) 1 + 2 * 3 *
5 3
4 5
-1 3 1
5 5 3
4 6 5
Row number 1
Column number 3
Element Row 1 and column 3
have been deleted
( -1) 1 + 3 * 1 *
5 5
4 6
Products are summed. If the element is zero then product is zero too.
= ( -1) 1 + 1 * ( -1) * 5 3
6 5
+ ( -1) 1 + 2 * 3 * 5 3
4 5
+ ( -1) 1 + 3 * 1 * 5 5 =
4 6
= - 5 3
6 5
- 3 * 5 3
4 5
+ 5 5 =
4 6
= - ( 5 * 5 - 3 * 6 )
  - 3 * ( 5 * 5 - 3 * 4 )
  + ( 5 * 6 - 5 * 4 ) =
= - ( 25 - 18 )
  - 3 * ( 25 - 12 )
  + ( 30 - 20 ) =
= -7
  - 39
  + 10 =
= -36
3. Let's calculate the determinant A by expanding along the row 2.
det A = -1 3 1 =
5 5 3
4 6 5
Expand the determinant along the row 2.   more info
-1 3 1
5 5 3
4 6 5
Row number 2
Column number 1
Element Row 2 and column 1
have been deleted
( -1) 2 + 1 * 5 *
3 1
6 5
-1 3 1
5 5 3
4 6 5
Row number 2
Column number 2
Element Row 2 and column 2
have been deleted
( -1) 2 + 2 * 5 *
-1 1
4 5
-1 3 1
5 5 3
4 6 5
Row number 2
Column number 3
Element Row 2 and column 3
have been deleted
( -1) 2 + 3 * 3 *
-1 3
4 6
Products are summed. If the element is zero then product is zero too.
= ( -1) 2 + 1 * 5 * 3 1
6 5
+ ( -1) 2 + 2 * 5 * -1 1
4 5
+ ( -1) 2 + 3 * 3 * -1 3 =
4 6
= - 5 * 3 1
6 5
+ 5 * -1 1
4 5
- 3 * -1 3 =
4 6
= - 5 * ( 3 * 5 - 1 * 6 )
  + 5 * ( -1 * 5 - 1 * 4 )
  - 3 * ( -1 * 6 - 3 * 4 ) =
= - 5 * ( 15 - 6 )
  + 5 * ( -5 - 4 )
  - 3 * ( -6 - 12 ) =
= -45
  - 45
  + 54 =
= -36
4. Let's calculate the determinant A by expanding along the row 3.
det A = -1 3 1 =
5 5 3
4 6 5
Expand the determinant along the row 3.   more info
-1 3 1
5 5 3
4 6 5
Row number 3
Column number 1
Element Row 3 and column 1
have been deleted
( -1) 3 + 1 * 4 *
3 1
5 3
-1 3 1
5 5 3
4 6 5
Row number 3
Column number 2
Element Row 3 and column 2
have been deleted
( -1) 3 + 2 * 6 *
-1 1
5 3
-1 3 1
5 5 3
4 6 5
Row number 3
Column number 3
Element Row 3 and column 3
have been deleted
( -1) 3 + 3 * 5 *
-1 3
5 5
Products are summed. If the element is zero then product is zero too.
= ( -1) 3 + 1 * 4 * 3 1
5 3
+ ( -1) 3 + 2 * 6 * -1 1
5 3
+ ( -1) 3 + 3 * 5 * -1 3 =
5 5
= 4 * 3 1
5 3
- 6 * -1 1
5 3
+ 5 * -1 3 =
5 5
= 4 * ( 3 * 3 - 1 * 5 )
  - 6 * ( -1 * 3 - 1 * 5 )
  + 5 * ( -1 * 5 - 3 * 5 ) =
= 4 * ( 9 - 5 )
  - 6 * ( -3 - 5 )
  + 5 * ( -5 - 15 ) =
= 16
  + 48
  - 100 =
= -36
5. Let's calculate the determinant A by expanding along the column 1.
det A = -1 3 1 =
5 5 3
4 6 5
Expand the determinant along the column 1.   more info
-1 3 1
5 5 3
4 6 5
Row number 1
Column number 1
Element Row 1 and column 1
have been deleted
( -1) 1 + 1 * -1 *
5 3
6 5
-1 3 1
5 5 3
4 6 5
Row number 2
Column number 1
Element Row 2 and column 1
have been deleted
( -1) 2 + 1 * 5 *
3 1
6 5
-1 3 1
5 5 3
4 6 5
Row number 3
Column number 1
Element Row 3 and column 1
have been deleted
( -1) 3 + 1 * 4 *
3 1
5 3
Products are summed. If the element is zero then product is zero too.
= ( -1) 1 + 1 * ( -1) * 5 3
6 5
+ ( -1) 2 + 1 * 5 * 3 1
6 5
+ ( -1) 3 + 1 * 4 * 3 1 =
5 3
= - 5 3
6 5
- 5 * 3 1
6 5
+ 4 * 3 1 =
5 3
= - ( 5 * 5 - 3 * 6 )
  - 5 * ( 3 * 5 - 1 * 6 )
  + 4 * ( 3 * 3 - 1 * 5 ) =
= - ( 25 - 18 )
  - 5 * ( 15 - 6 )
  + 4 * ( 9 - 5 ) =
= -7
  - 45
  + 16 =
= -36
6. Let's calculate the determinant A by expanding along the column 2.
det A = -1 3 1 =
5 5 3
4 6 5
Expand the determinant along the column 2.   more info
-1 3 1
5 5 3
4 6 5
Row number 1
Column number 2
Element Row 1 and column 2
have been deleted
( -1) 1 + 2 * 3 *
5 3
4 5
-1 3 1
5 5 3
4 6 5
Row number 2
Column number 2
Element Row 2 and column 2
have been deleted
( -1) 2 + 2 * 5 *
-1 1
4 5
-1 3 1
5 5 3
4 6 5
Row number 3
Column number 2
Element Row 3 and column 2
have been deleted
( -1) 3 + 2 * 6 *
-1 1
5 3
Products are summed. If the element is zero then product is zero too.
= ( -1) 1 + 2 * 3 * 5 3
4 5
+ ( -1) 2 + 2 * 5 * -1 1
4 5
+ ( -1) 3 + 2 * 6 * -1 1 =
5 3
= - 3 * 5 3
4 5
+ 5 * -1 1
4 5
- 6 * -1 1 =
5 3
= - 3 * ( 5 * 5 - 3 * 4 )
  + 5 * ( -1 * 5 - 1 * 4 )
  - 6 * ( -1 * 3 - 1 * 5 ) =
= - 3 * ( 25 - 12 )
  + 5 * ( -5 - 4 )
  - 6 * ( -3 - 5 ) =
= -39
  - 45
  + 48 =
= -36
7. Let's calculate the determinant A by expanding along the column 3.
det A = -1 3 1 =
5 5 3
4 6 5
Expand the determinant along the column 3.   more info
-1 3 1
5 5 3
4 6 5
Row number 1
Column number 3
Element Row 1 and column 3
have been deleted
( -1) 1 + 3 * 1 *
5 5
4 6
-1 3 1
5 5 3
4 6 5
Row number 2
Column number 3
Element Row 2 and column 3
have been deleted
( -1) 2 + 3 * 3 *
-1 3
4 6
-1 3 1
5 5 3
4 6 5
Row number 3
Column number 3
Element Row 3 and column 3
have been deleted
( -1) 3 + 3 * 5 *
-1 3
5 5
Products are summed. If the element is zero then product is zero too.
= ( -1) 1 + 3 * 1 * 5 5
4 6
+ ( -1) 2 + 3 * 3 * -1 3
4 6
+ ( -1) 3 + 3 * 5 * -1 3 =
5 5
= 5 5
4 6
- 3 * -1 3
4 6
+ 5 * -1 3 =
5 5
= ( 5 * 6 - 5 * 4 )
  - 3 * ( -1 * 6 - 3 * 4 )
  + 5 * ( -1 * 5 - 3 * 5 ) =
= ( 30 - 20 )
  - 3 * ( -6 - 12 )
  + 5 * ( -5 - 15 ) =
= 10
  + 54
  - 100 =
= -36








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