﻿ Determinant of a 3x3 Matrix Calculator

# Example of Finding the Determinant of a 3x3 Matrix

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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
 -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
 -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
 -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
 -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
 -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
 -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
 -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