Example of Finding the Determinant of a 3x3 Matrix

This solution has been done by the calculator presented on the site.

1. Let's calculate the determinant A using a elementary transformations.
det A = 1 -1 -4 =
3 3 3
-2 6 4
The elements of column 1 multiplied by -1 are added to the corresponding elements of column 2.   more info
1 -1 + 1 * ( -1) -4
3 3 + 3 * ( -1) 3
-2 6 + ( -2) * ( -1) 4
This elementary transformation does not change the value of the determinant.
= 1 -2 -4 =
3 0 3
-2 8 4
The elements of column 1 multiplied by -1 are added to the corresponding elements of column 3.   more info
1 -2 -4 + 1 * ( -1)
3 0 3 + 3 * ( -1)
-2 8 4 + ( -2) * ( -1)
This elementary transformation does not change the value of the determinant.
= 1 -2 -5 =
3 0 0
-2 8 6
Expand the determinant along the row 2.   more info
1 -2 -5
3 0 0
-2 8 6
Row number 2
Column number 1
Element Row 2 and column 1
have been deleted
( -1) 2 + 1 * 3 *
-2 -5
8 6
1 -2 -5
3 0 0
-2 8 6
Row number 2
Column number 2
Element Row 2 and column 2
have been deleted
( -1) 2 + 2 * 0 *
1 -5
-2 6
1 -2 -5
3 0 0
-2 8 6
Row number 2
Column number 3
Element Row 2 and column 3
have been deleted
( -1) 2 + 3 * 0 *
1 -2
-2 8
Products are summed. If the element is zero than product is zero too.
= ( -1) 2 + 1 * 3 * -2 -5 =
8 6
= - 3 * -2 -5 =
8 6
= - 3 * ( -2 * 6 - ( -5) * 8 ) =
= - 3 * ( -12 + 40 ) =
= -84
2. Let's calculate the determinant A by expanding along the row 1.
det A = 1 -1 -4 =
3 3 3
-2 6 4
Expand the determinant along the row 1.   more info
1 -1 -4
3 3 3
-2 6 4
Row number 1
Column number 1
Element Row 1 and column 1
have been deleted
( -1) 1 + 1 * 1 *
3 3
6 4
1 -1 -4
3 3 3
-2 6 4
Row number 1
Column number 2
Element Row 1 and column 2
have been deleted
( -1) 1 + 2 * -1 *
3 3
-2 4
1 -1 -4
3 3 3
-2 6 4
Row number 1
Column number 3
Element Row 1 and column 3
have been deleted
( -1) 1 + 3 * -4 *
3 3
-2 6
Products are summed. If the element is zero than product is zero too.
= ( -1) 1 + 1 * 1 * 3 3
6 4
+ ( -1) 1 + 2 * ( -1) * 3 3
-2 4
+ ( -1) 1 + 3 * ( -4) * 3 3 =
-2 6
= 3 3
6 4
+ 3 3
-2 4
- 4 * 3 3 =
-2 6
= ( 3 * 4 - 3 * 6 )
  + ( 3 * 4 - 3 * ( -2) )
  - 4 * ( 3 * 6 - 3 * ( -2) ) =
= ( 12 - 18 )
  + ( 12 + 6 )
  - 4 * ( 18 + 6 ) =
= -6
  + 18
  - 96 =
= -84
3. Let's calculate the determinant A by expanding along the row 2.
det A = 1 -1 -4 =
3 3 3
-2 6 4
Expand the determinant along the row 2.   more info
1 -1 -4
3 3 3
-2 6 4
Row number 2
Column number 1
Element Row 2 and column 1
have been deleted
( -1) 2 + 1 * 3 *
-1 -4
6 4
1 -1 -4
3 3 3
-2 6 4
Row number 2
Column number 2
Element Row 2 and column 2
have been deleted
( -1) 2 + 2 * 3 *
1 -4
-2 4
1 -1 -4
3 3 3
-2 6 4
Row number 2
Column number 3
Element Row 2 and column 3
have been deleted
( -1) 2 + 3 * 3 *
1 -1
-2 6
Products are summed. If the element is zero than product is zero too.
= ( -1) 2 + 1 * 3 * -1 -4
6 4
+ ( -1) 2 + 2 * 3 * 1 -4
-2 4
+ ( -1) 2 + 3 * 3 * 1 -1 =
-2 6
= - 3 * -1 -4
6 4
+ 3 * 1 -4
-2 4
- 3 * 1 -1 =
-2 6
= - 3 * ( -1 * 4 - ( -4) * 6 )
  + 3 * ( 1 * 4 - ( -4) * ( -2) )
  - 3 * ( 1 * 6 - ( -1) * ( -2) ) =
= - 3 * ( -4 + 24 )
  + 3 * ( 4 - 8 )
  - 3 * ( 6 - 2 ) =
= -60
  - 12
  - 12 =
= -84
4. Let's calculate the determinant A by expanding along the row 3.
det A = 1 -1 -4 =
3 3 3
-2 6 4
Expand the determinant along the row 3.   more info
1 -1 -4
3 3 3
-2 6 4
Row number 3
Column number 1
Element Row 3 and column 1
have been deleted
( -1) 3 + 1 * -2 *
-1 -4
3 3
1 -1 -4
3 3 3
-2 6 4
Row number 3
Column number 2
Element Row 3 and column 2
have been deleted
( -1) 3 + 2 * 6 *
1 -4
3 3
1 -1 -4
3 3 3
-2 6 4
Row number 3
Column number 3
Element Row 3 and column 3
have been deleted
( -1) 3 + 3 * 4 *
1 -1
3 3
Products are summed. If the element is zero than product is zero too.
= ( -1) 3 + 1 * ( -2) * -1 -4
3 3
+ ( -1) 3 + 2 * 6 * 1 -4
3 3
+ ( -1) 3 + 3 * 4 * 1 -1 =
3 3
= - 2 * -1 -4
3 3
- 6 * 1 -4
3 3
+ 4 * 1 -1 =
3 3
= - 2 * ( -1 * 3 - ( -4) * 3 )
  - 6 * ( 1 * 3 - ( -4) * 3 )
  + 4 * ( 1 * 3 - ( -1) * 3 ) =
= - 2 * ( -3 + 12 )
  - 6 * ( 3 + 12 )
  + 4 * ( 3 + 3 ) =
= -18
  - 90
  + 24 =
= -84
5. Let's calculate the determinant A by expanding along the column 1.
det A = 1 -1 -4 =
3 3 3
-2 6 4
Expand the determinant along the column 1.   more info
1 -1 -4
3 3 3
-2 6 4
Row number 1
Column number 1
Element Row 1 and column 1
have been deleted
( -1) 1 + 1 * 1 *
3 3
6 4
1 -1 -4
3 3 3
-2 6 4
Row number 2
Column number 1
Element Row 2 and column 1
have been deleted
( -1) 2 + 1 * 3 *
-1 -4
6 4
1 -1 -4
3 3 3
-2 6 4
Row number 3
Column number 1
Element Row 3 and column 1
have been deleted
( -1) 3 + 1 * -2 *
-1 -4
3 3
Products are summed. If the element is zero than product is zero too.
= ( -1) 1 + 1 * 1 * 3 3
6 4
+ ( -1) 2 + 1 * 3 * -1 -4
6 4
+ ( -1) 3 + 1 * ( -2) * -1 -4 =
3 3
= 3 3
6 4
- 3 * -1 -4
6 4
- 2 * -1 -4 =
3 3
= ( 3 * 4 - 3 * 6 )
  - 3 * ( -1 * 4 - ( -4) * 6 )
  - 2 * ( -1 * 3 - ( -4) * 3 ) =
= ( 12 - 18 )
  - 3 * ( -4 + 24 )
  - 2 * ( -3 + 12 ) =
= -6
  - 60
  - 18 =
= -84
6. Let's calculate the determinant A by expanding along the column 2.
det A = 1 -1 -4 =
3 3 3
-2 6 4
Expand the determinant along the column 2.   more info
1 -1 -4
3 3 3
-2 6 4
Row number 1
Column number 2
Element Row 1 and column 2
have been deleted
( -1) 1 + 2 * -1 *
3 3
-2 4
1 -1 -4
3 3 3
-2 6 4
Row number 2
Column number 2
Element Row 2 and column 2
have been deleted
( -1) 2 + 2 * 3 *
1 -4
-2 4
1 -1 -4
3 3 3
-2 6 4
Row number 3
Column number 2
Element Row 3 and column 2
have been deleted
( -1) 3 + 2 * 6 *
1 -4
3 3
Products are summed. If the element is zero than product is zero too.
= ( -1) 1 + 2 * ( -1) * 3 3
-2 4
+ ( -1) 2 + 2 * 3 * 1 -4
-2 4
+ ( -1) 3 + 2 * 6 * 1 -4 =
3 3
= 3 3
-2 4
+ 3 * 1 -4
-2 4
- 6 * 1 -4 =
3 3
= ( 3 * 4 - 3 * ( -2) )
  + 3 * ( 1 * 4 - ( -4) * ( -2) )
  - 6 * ( 1 * 3 - ( -4) * 3 ) =
= ( 12 + 6 )
  + 3 * ( 4 - 8 )
  - 6 * ( 3 + 12 ) =
= 18
  - 12
  - 90 =
= -84
7. Let's calculate the determinant A by expanding along the column 3.
det A = 1 -1 -4 =
3 3 3
-2 6 4
Expand the determinant along the column 3.   more info
1 -1 -4
3 3 3
-2 6 4
Row number 1
Column number 3
Element Row 1 and column 3
have been deleted
( -1) 1 + 3 * -4 *
3 3
-2 6
1 -1 -4
3 3 3
-2 6 4
Row number 2
Column number 3
Element Row 2 and column 3
have been deleted
( -1) 2 + 3 * 3 *
1 -1
-2 6
1 -1 -4
3 3 3
-2 6 4
Row number 3
Column number 3
Element Row 3 and column 3
have been deleted
( -1) 3 + 3 * 4 *
1 -1
3 3
Products are summed. If the element is zero than product is zero too.
= ( -1) 1 + 3 * ( -4) * 3 3
-2 6
+ ( -1) 2 + 3 * 3 * 1 -1
-2 6
+ ( -1) 3 + 3 * 4 * 1 -1 =
3 3
= - 4 * 3 3
-2 6
- 3 * 1 -1
-2 6
+ 4 * 1 -1 =
3 3
= - 4 * ( 3 * 6 - 3 * ( -2) )
  - 3 * ( 1 * 6 - ( -1) * ( -2) )
  + 4 * ( 1 * 3 - ( -1) * 3 ) =
= - 4 * ( 18 + 6 )
  - 3 * ( 6 - 2 )
  + 4 * ( 3 + 3 ) =
= -96
  - 12
  + 24 =
= -84
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