Example of Finding the Determinant of a 4x4 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 = 
 3  3  5  8   = 
3  2  4  6 
2  5  7  5 
4  3  5  6 
The elements of row 2 multiplied by 1 are added to the corresponding elements of row 4.
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 3  3  5  8  
3  2  4  6 
2  5  7  5 
4 + ( 3) * ( 1)  3 + 2 * ( 1)  5 + 4 * ( 1)  6 + ( 6) * ( 1) 
This elementary transformation does not change the value of the determinant.
= 
 3  3  5  8   = 
3  2  4  6 
2  5  7  5 
1  1  1  0 
The elements of column 1 are added to the corresponding elements of column 2.
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 3  3 + 3  5  8  
3  2 + ( 3)  4  6 
2  5 + 2  7  5 
1  1 + ( 1)  1  0 
This elementary transformation does not change the value of the determinant.
= 
 3  0  5  8   = 
3  1  4  6 
2  3  7  5 
1  0  1  0 
The elements of column 1 are added to the corresponding elements of column 3.
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 3  0  5 + 3  8  
3  1  4 + ( 3)  6 
2  3  7 + 2  5 
1  0  1 + ( 1)  0 
This elementary transformation does not change the value of the determinant.
= 
 3  0  2  8   = 
3  1  1  6 
2  3  5  5 
1  0  0  0 
Expand the determinant along the row 4.
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 3  0  2  8   3  1  1  6  2  3  5  5  1  0  0  0  
Row number 4 Column number 1 

Element 

Row 4 and column 1 have been deleted 
( 1) ^{4 + 1} 
* 
1 
* 

 3  0  2  8   3  1  1  6  2  3  5  5  1  0  0  0  
Row number 4 Column number 2 

Element 

Row 4 and column 2 have been deleted 
( 1) ^{4 + 2} 
* 
0 
* 

 3  0  2  8   3  1  1  6  2  3  5  5  1  0  0  0  
Row number 4 Column number 3 

Element 

Row 4 and column 3 have been deleted 
( 1) ^{4 + 3} 
* 
0 
* 

 3  0  2  8   3  1  1  6  2  3  5  5  1  0  0  0  
Row number 4 Column number 4 

Element 

Row 4 and column 4 have been deleted 
( 1) ^{4 + 4} 
* 
0 
* 

Products are summed. If the element is zero than product is zero too.
= ( 1) ^{4 + 1} * ( 1) * 
 0  2  8   = 
1  1  6 
3  5  5 
= 
 0  2  8   = 
1  1  6 
3  5  5 
The elements of row 2 multiplied by 3 are added to the corresponding elements of row 3.
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 0  2  8  
1  1  6 
3 + ( 1) * ( 3)  5 + 1 * ( 3)  5 + ( 6) * ( 3) 
This elementary transformation does not change the value of the determinant.
= 
 0  2  8   = 
1  1  6 
0  8  23 
Expand the determinant along the column 1.
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Row number 1 Column number 1 

Element 

Row 1 and column 1 have been deleted 
( 1) ^{1 + 1} 
* 
0 
* 


Row number 2 Column number 1 

Element 

Row 2 and column 1 have been deleted 
( 1) ^{2 + 1} 
* 
1 
* 


Row number 3 Column number 1 

Element 

Row 3 and column 1 have been deleted 
( 1) ^{3 + 1} 
* 
0 
* 

Products are summed. If the element is zero than product is zero too.
= ( 1) ^{2 + 1} * ( 1) * 
 2  8   = 
8  23 
= 18