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Example ¹2. Finding the Determinant of a 4x4 Matrix

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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
Expand the determinant along the column 4.   more info
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
The elements of column 2 are added to the corresponding elements of column 3.   more info
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
Expand the determinant along the row 1.   more info
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

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