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Example 3. Finding the Determinant of a 5x5 Matrix

This solution has been made using the calculator presented on the site.
Let's calculate the determinant A using a elementary transformations.
det A = 4 1 1 2 1 =
1 2 -1 1 1
3 1 1 1 1
2 1 1 4 1
2 -1 1 1 5
The elements of row 3 multiplied by -1 are added to the corresponding elements of row 1.   more info
4 + 3 * ( -1) 1 + 1 * ( -1) 1 + 1 * ( -1) 2 + 1 * ( -1) 1 + 1 * ( -1)
1 2 -1 1 1
3 1 1 1 1
2 1 1 4 1
2 -1 1 1 5
This elementary transformation does not change the value of the determinant.
= 1 0 0 1 0 =
1 2 -1 1 1
3 1 1 1 1
2 1 1 4 1
2 -1 1 1 5
The elements of column 4 multiplied by -1 are added to the corresponding elements of column 1.   more info
1 + 1 * ( -1) 0 0 1 0
1 + 1 * ( -1) 2 -1 1 1
3 + 1 * ( -1) 1 1 1 1
2 + 4 * ( -1) 1 1 4 1
2 + 1 * ( -1) -1 1 1 5
This elementary transformation does not change the value of the determinant.
= 0 0 0 1 0 =
0 2 -1 1 1
2 1 1 1 1
-2 1 1 4 1
1 -1 1 1 5
Expand the determinant along the row 1.   more info
0 0 0 1 0
0 2 -1 1 1
2 1 1 1 1
-2 1 1 4 1
1 -1 1 1 5
Row number 1
Column number 1
Element Row 1 and column 1
have been deleted
( -1) 1 + 1 * 0 *
2 -1 1 1
1 1 1 1
1 1 4 1
-1 1 1 5
0 0 0 1 0
0 2 -1 1 1
2 1 1 1 1
-2 1 1 4 1
1 -1 1 1 5
Row number 1
Column number 2
Element Row 1 and column 2
have been deleted
( -1) 1 + 2 * 0 *
0 -1 1 1
2 1 1 1
-2 1 4 1
1 1 1 5
0 0 0 1 0
0 2 -1 1 1
2 1 1 1 1
-2 1 1 4 1
1 -1 1 1 5
Row number 1
Column number 3
Element Row 1 and column 3
have been deleted
( -1) 1 + 3 * 0 *
0 2 1 1
2 1 1 1
-2 1 4 1
1 -1 1 5
0 0 0 1 0
0 2 -1 1 1
2 1 1 1 1
-2 1 1 4 1
1 -1 1 1 5
Row number 1
Column number 4
Element Row 1 and column 4
have been deleted
( -1) 1 + 4 * 1 *
0 2 -1 1
2 1 1 1
-2 1 1 1
1 -1 1 5
0 0 0 1 0
0 2 -1 1 1
2 1 1 1 1
-2 1 1 4 1
1 -1 1 1 5
Row number 1
Column number 5
Element Row 1 and column 5
have been deleted
( -1) 1 + 5 * 0 *
0 2 -1 1
2 1 1 1
-2 1 1 4
1 -1 1 1
Products are summed. If the element is zero then product is zero too.
= ( -1) 1 + 4 * 1 * 0 2 -1 1 =
2 1 1 1
-2 1 1 1
1 -1 1 5
= - 0 2 -1 1 =
2 1 1 1
-2 1 1 1
1 -1 1 5
The elements of row 2 multiplied by -1 are added to the corresponding elements of row 3.   more info
0 2 -1 1
2 1 1 1
-2 + 2 * ( -1) 1 + 1 * ( -1) 1 + 1 * ( -1) 1 + 1 * ( -1)
1 -1 1 5
This elementary transformation does not change the value of the determinant.
= - 0 2 -1 1 =
2 1 1 1
-4 0 0 0
1 -1 1 5
Expand the determinant along the row 3.   more info
0 2 -1 1
2 1 1 1
-4 0 0 0
1 -1 1 5
Row number 3
Column number 1
Element Row 3 and column 1
have been deleted
( -1) 3 + 1 * -4 *
2 -1 1
1 1 1
-1 1 5
0 2 -1 1
2 1 1 1
-4 0 0 0
1 -1 1 5
Row number 3
Column number 2
Element Row 3 and column 2
have been deleted
( -1) 3 + 2 * 0 *
0 -1 1
2 1 1
1 1 5
0 2 -1 1
2 1 1 1
-4 0 0 0
1 -1 1 5
Row number 3
Column number 3
Element Row 3 and column 3
have been deleted
( -1) 3 + 3 * 0 *
0 2 1
2 1 1
1 -1 5
0 2 -1 1
2 1 1 1
-4 0 0 0
1 -1 1 5
Row number 3
Column number 4
Element Row 3 and column 4
have been deleted
( -1) 3 + 4 * 0 *
0 2 -1
2 1 1
1 -1 1
Products are summed. If the element is zero then product is zero too.
= - ( ( -1) 3 + 1 * ( -4) * 2 -1 1 ) =
1 1 1
-1 1 5
= 4 * 2 -1 1 =
1 1 1
-1 1 5
The elements of row 2 multiplied by -1 are added to the corresponding elements of row 3.   more info
2 -1 1
1 1 1
-1 + 1 * ( -1) 1 + 1 * ( -1) 5 + 1 * ( -1)
This elementary transformation does not change the value of the determinant.
= 4 * 2 -1 1 =
1 1 1
-2 0 4
The elements of row 2 are added to the corresponding elements of row 1.   more info
2 + 1 -1 + 1 1 + 1
1 1 1
-2 0 4
This elementary transformation does not change the value of the determinant.
= 4 * 3 0 2 =
1 1 1
-2 0 4
Expand the determinant along the column 2.   more info
3 0 2
1 1 1
-2 0 4
Row number 1
Column number 2
Element Row 1 and column 2
have been deleted
( -1) 1 + 2 * 0 *
1 1
-2 4
3 0 2
1 1 1
-2 0 4
Row number 2
Column number 2
Element Row 2 and column 2
have been deleted
( -1) 2 + 2 * 1 *
3 2
-2 4
3 0 2
1 1 1
-2 0 4
Row number 3
Column number 2
Element Row 3 and column 2
have been deleted
( -1) 3 + 2 * 0 *
3 2
1 1
Products are summed. If the element is zero then product is zero too.
= 4 * ( -1) 2 + 2 * 1 * 3 2 =
-2 4
= 4 * 3 2 =
-2 4
= 4 * ( 3 * 4 - 2 * ( -2) ) =
= 4 * ( 12 + 4 ) =
= 64








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