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 = -3 -3 -4 =
-4 4 -6
4 -6 5
The elements of column 1 multiplied by -1 are added to the corresponding elements of column 3.   more info
-3 -3 -4 + ( -3) * ( -1)
-4 4 -6 + ( -4) * ( -1)
4 -6 5 + 4 * ( -1)
This elementary transformation does not change the value of the determinant.
= -3 -3 -1 =
-4 4 -2
4 -6 1
The elements of column 3 multiplied by -2 are added to the corresponding elements of column 1.   more info
-3 + ( -1) * ( -2) -3 -1
-4 + ( -2) * ( -2) 4 -2
4 + 1 * ( -2) -6 1
This elementary transformation does not change the value of the determinant.
= -1 -3 -1 =
0 4 -2
2 -6 1
The elements of column 3 multiplied by 2 are added to the corresponding elements of column 2.   more info
-1 -3 + ( -1) * 2 -1
0 4 + ( -2) * 2 -2
2 -6 + 1 * 2 1
This elementary transformation does not change the value of the determinant.
= -1 -5 -1 =
0 0 -2
2 -4 1
Expand the determinant along the row 2.   more info
-1 -5 -1
0 0 -2
2 -4 1
Row number 2
Column number 1
Element Row 2 and column 1
have been deleted
( -1) 2 + 1 * 0 *
-5 -1
-4 1
-1 -5 -1
0 0 -2
2 -4 1
Row number 2
Column number 2
Element Row 2 and column 2
have been deleted
( -1) 2 + 2 * 0 *
-1 -1
2 1
-1 -5 -1
0 0 -2
2 -4 1
Row number 2
Column number 3
Element Row 2 and column 3
have been deleted
( -1) 2 + 3 * -2 *
-1 -5
2 -4
Products are summed. If the element is zero than product is zero too.
= ( -1) 2 + 3 * ( -2) * -1 -5 =
2 -4
= 2 * -1 -5 =
2 -4
= 2 * ( -1 * ( -4) - ( -5) * 2 ) =
= 2 * ( 4 + 10 ) =
= 28
2. Let's calculate the determinant A by expanding along the row 1.
det A = -3 -3 -4 =
-4 4 -6
4 -6 5
Expand the determinant along the row 1.   more info
-3 -3 -4
-4 4 -6
4 -6 5
Row number 1
Column number 1
Element Row 1 and column 1
have been deleted
( -1) 1 + 1 * -3 *
4 -6
-6 5
-3 -3 -4
-4 4 -6
4 -6 5
Row number 1
Column number 2
Element Row 1 and column 2
have been deleted
( -1) 1 + 2 * -3 *
-4 -6
4 5
-3 -3 -4
-4 4 -6
4 -6 5
Row number 1
Column number 3
Element Row 1 and column 3
have been deleted
( -1) 1 + 3 * -4 *
-4 4
4 -6
Products are summed. If the element is zero than product is zero too.
= ( -1) 1 + 1 * ( -3) * 4 -6
-6 5
+ ( -1) 1 + 2 * ( -3) * -4 -6
4 5
+ ( -1) 1 + 3 * ( -4) * -4 4 =
4 -6
= - 3 * 4 -6
-6 5
+ 3 * -4 -6
4 5
- 4 * -4 4 =
4 -6
= - 3 * ( 4 * 5 - ( -6) * ( -6) )
  + 3 * ( -4 * 5 - ( -6) * 4 )
  - 4 * ( -4 * ( -6) - 4 * 4 ) =
= - 3 * ( 20 - 36 )
  + 3 * ( -20 + 24 )
  - 4 * ( 24 - 16 ) =
= 48
  + 12
  - 32 =
= 28
3. Let's calculate the determinant A by expanding along the row 2.
det A = -3 -3 -4 =
-4 4 -6
4 -6 5
Expand the determinant along the row 2.   more info
-3 -3 -4
-4 4 -6
4 -6 5
Row number 2
Column number 1
Element Row 2 and column 1
have been deleted
( -1) 2 + 1 * -4 *
-3 -4
-6 5
-3 -3 -4
-4 4 -6
4 -6 5
Row number 2
Column number 2
Element Row 2 and column 2
have been deleted
( -1) 2 + 2 * 4 *
-3 -4
4 5
-3 -3 -4
-4 4 -6
4 -6 5
Row number 2
Column number 3
Element Row 2 and column 3
have been deleted
( -1) 2 + 3 * -6 *
-3 -3
4 -6
Products are summed. If the element is zero than product is zero too.
= ( -1) 2 + 1 * ( -4) * -3 -4
-6 5
+ ( -1) 2 + 2 * 4 * -3 -4
4 5
+ ( -1) 2 + 3 * ( -6) * -3 -3 =
4 -6
= 4 * -3 -4
-6 5
+ 4 * -3 -4
4 5
+ 6 * -3 -3 =
4 -6
= 4 * ( -3 * 5 - ( -4) * ( -6) )
  + 4 * ( -3 * 5 - ( -4) * 4 )
  + 6 * ( -3 * ( -6) - ( -3) * 4 ) =
= 4 * ( -15 - 24 )
  + 4 * ( -15 + 16 )
  + 6 * ( 18 + 12 ) =
= -156
  + 4
  + 180 =
= 28
4. Let's calculate the determinant A by expanding along the row 3.
det A = -3 -3 -4 =
-4 4 -6
4 -6 5
Expand the determinant along the row 3.   more info
-3 -3 -4
-4 4 -6
4 -6 5
Row number 3
Column number 1
Element Row 3 and column 1
have been deleted
( -1) 3 + 1 * 4 *
-3 -4
4 -6
-3 -3 -4
-4 4 -6
4 -6 5
Row number 3
Column number 2
Element Row 3 and column 2
have been deleted
( -1) 3 + 2 * -6 *
-3 -4
-4 -6
-3 -3 -4
-4 4 -6
4 -6 5
Row number 3
Column number 3
Element Row 3 and column 3
have been deleted
( -1) 3 + 3 * 5 *
-3 -3
-4 4
Products are summed. If the element is zero than product is zero too.
= ( -1) 3 + 1 * 4 * -3 -4
4 -6
+ ( -1) 3 + 2 * ( -6) * -3 -4
-4 -6
+ ( -1) 3 + 3 * 5 * -3 -3 =
-4 4
= 4 * -3 -4
4 -6
+ 6 * -3 -4
-4 -6
+ 5 * -3 -3 =
-4 4
= 4 * ( -3 * ( -6) - ( -4) * 4 )
  + 6 * ( -3 * ( -6) - ( -4) * ( -4) )
  + 5 * ( -3 * 4 - ( -3) * ( -4) ) =
= 4 * ( 18 + 16 )
  + 6 * ( 18 - 16 )
  + 5 * ( -12 - 12 ) =
= 136
  + 12
  - 120 =
= 28
5. Let's calculate the determinant A by expanding along the column 1.
det A = -3 -3 -4 =
-4 4 -6
4 -6 5
Expand the determinant along the column 1.   more info
-3 -3 -4
-4 4 -6
4 -6 5
Row number 1
Column number 1
Element Row 1 and column 1
have been deleted
( -1) 1 + 1 * -3 *
4 -6
-6 5
-3 -3 -4
-4 4 -6
4 -6 5
Row number 2
Column number 1
Element Row 2 and column 1
have been deleted
( -1) 2 + 1 * -4 *
-3 -4
-6 5
-3 -3 -4
-4 4 -6
4 -6 5
Row number 3
Column number 1
Element Row 3 and column 1
have been deleted
( -1) 3 + 1 * 4 *
-3 -4
4 -6
Products are summed. If the element is zero than product is zero too.
= ( -1) 1 + 1 * ( -3) * 4 -6
-6 5
+ ( -1) 2 + 1 * ( -4) * -3 -4
-6 5
+ ( -1) 3 + 1 * 4 * -3 -4 =
4 -6
= - 3 * 4 -6
-6 5
+ 4 * -3 -4
-6 5
+ 4 * -3 -4 =
4 -6
= - 3 * ( 4 * 5 - ( -6) * ( -6) )
  + 4 * ( -3 * 5 - ( -4) * ( -6) )
  + 4 * ( -3 * ( -6) - ( -4) * 4 ) =
= - 3 * ( 20 - 36 )
  + 4 * ( -15 - 24 )
  + 4 * ( 18 + 16 ) =
= 48
  - 156
  + 136 =
= 28
6. Let's calculate the determinant A by expanding along the column 2.
det A = -3 -3 -4 =
-4 4 -6
4 -6 5
Expand the determinant along the column 2.   more info
-3 -3 -4
-4 4 -6
4 -6 5
Row number 1
Column number 2
Element Row 1 and column 2
have been deleted
( -1) 1 + 2 * -3 *
-4 -6
4 5
-3 -3 -4
-4 4 -6
4 -6 5
Row number 2
Column number 2
Element Row 2 and column 2
have been deleted
( -1) 2 + 2 * 4 *
-3 -4
4 5
-3 -3 -4
-4 4 -6
4 -6 5
Row number 3
Column number 2
Element Row 3 and column 2
have been deleted
( -1) 3 + 2 * -6 *
-3 -4
-4 -6
Products are summed. If the element is zero than product is zero too.
= ( -1) 1 + 2 * ( -3) * -4 -6
4 5
+ ( -1) 2 + 2 * 4 * -3 -4
4 5
+ ( -1) 3 + 2 * ( -6) * -3 -4 =
-4 -6
= 3 * -4 -6
4 5
+ 4 * -3 -4
4 5
+ 6 * -3 -4 =
-4 -6
= 3 * ( -4 * 5 - ( -6) * 4 )
  + 4 * ( -3 * 5 - ( -4) * 4 )
  + 6 * ( -3 * ( -6) - ( -4) * ( -4) ) =
= 3 * ( -20 + 24 )
  + 4 * ( -15 + 16 )
  + 6 * ( 18 - 16 ) =
= 12
  + 4
  + 12 =
= 28
7. Let's calculate the determinant A by expanding along the column 3.
det A = -3 -3 -4 =
-4 4 -6
4 -6 5
Expand the determinant along the column 3.   more info
-3 -3 -4
-4 4 -6
4 -6 5
Row number 1
Column number 3
Element Row 1 and column 3
have been deleted
( -1) 1 + 3 * -4 *
-4 4
4 -6
-3 -3 -4
-4 4 -6
4 -6 5
Row number 2
Column number 3
Element Row 2 and column 3
have been deleted
( -1) 2 + 3 * -6 *
-3 -3
4 -6
-3 -3 -4
-4 4 -6
4 -6 5
Row number 3
Column number 3
Element Row 3 and column 3
have been deleted
( -1) 3 + 3 * 5 *
-3 -3
-4 4
Products are summed. If the element is zero than product is zero too.
= ( -1) 1 + 3 * ( -4) * -4 4
4 -6
+ ( -1) 2 + 3 * ( -6) * -3 -3
4 -6
+ ( -1) 3 + 3 * 5 * -3 -3 =
-4 4
= - 4 * -4 4
4 -6
+ 6 * -3 -3
4 -6
+ 5 * -3 -3 =
-4 4
= - 4 * ( -4 * ( -6) - 4 * 4 )
  + 6 * ( -3 * ( -6) - ( -3) * 4 )
  + 5 * ( -3 * 4 - ( -3) * ( -4) ) =
= - 4 * ( 24 - 16 )
  + 6 * ( 18 + 12 )
  + 5 * ( -12 - 12 ) =
= -32
  + 180
  - 120 =
= 28
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