Example ¹1. Finding the Determinant of a 3x3 MatrixThis solution was made using the calculator presented on the site. Example ¹2. Finding the determinant of a 4x4 matrix Example ¹3. Finding the determinant of a 5x5 matrix 1. Let's calculate the determinant A using a elementary transformations.
The elements of row 1 multiplied by 2 are added to the corresponding elements of row 3. more info
This elementary transformation does not change the value of the determinant.
The elements of column 3 multiplied by 2 are added to the corresponding elements of column 1. more info
This elementary transformation does not change the value of the determinant.
Expand the determinant along the row 3. more info
Products are summed. If the element is zero then product is zero too.
= 3 * ( 3 * 5  3 * ( 1) ) = = 3 * ( 15 + 3 ) = = 36 2. Let's calculate the determinant A by expanding along the row 1.
Expand the determinant along the row 1. more info
Products are summed. If the element is zero then product is zero too.
=  ( 5 * 5  3 * 6 )  3 * ( 5 * 5  3 * 4 ) + ( 5 * 6  5 * 4 ) = =  ( 25  18 )  3 * ( 25  12 ) + ( 30  20 ) = = 7  39 + 10 = = 36 3. Let's calculate the determinant A by expanding along the row 2.
Expand the determinant along the row 2. more info
Products are summed. If the element is zero then product is zero too.
=  5 * ( 3 * 5  1 * 6 ) + 5 * ( 1 * 5  1 * 4 )  3 * ( 1 * 6  3 * 4 ) = =  5 * ( 15  6 ) + 5 * ( 5  4 )  3 * ( 6  12 ) = = 45  45 + 54 = = 36 4. Let's calculate the determinant A by expanding along the row 3.
Expand the determinant along the row 3. more info
Products are summed. If the element is zero then product is zero too.
= 4 * ( 3 * 3  1 * 5 )  6 * ( 1 * 3  1 * 5 ) + 5 * ( 1 * 5  3 * 5 ) = = 4 * ( 9  5 )  6 * ( 3  5 ) + 5 * ( 5  15 ) = = 16 + 48  100 = = 36 5. Let's calculate the determinant A by expanding along the column 1.
Expand the determinant along the column 1. more info
Products are summed. If the element is zero then product is zero too.
=  ( 5 * 5  3 * 6 )  5 * ( 3 * 5  1 * 6 ) + 4 * ( 3 * 3  1 * 5 ) = =  ( 25  18 )  5 * ( 15  6 ) + 4 * ( 9  5 ) = = 7  45 + 16 = = 36 6. Let's calculate the determinant A by expanding along the column 2.
Expand the determinant along the column 2. more info
Products are summed. If the element is zero then product is zero too.
=  3 * ( 5 * 5  3 * 4 ) + 5 * ( 1 * 5  1 * 4 )  6 * ( 1 * 3  1 * 5 ) = =  3 * ( 25  12 ) + 5 * ( 5  4 )  6 * ( 3  5 ) = = 39  45 + 48 = = 36 7. Let's calculate the determinant A by expanding along the column 3.
Expand the determinant along the column 3. more info
Products are summed. If the element is zero then product is zero too.
= ( 5 * 6  5 * 4 )  3 * ( 1 * 6  3 * 4 ) + 5 * ( 1 * 5  3 * 5 ) = = ( 30  20 )  3 * ( 6  12 ) + 5 * ( 5  15 ) = = 10 + 54  100 = = 36
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