Dimension matrix:
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Method:
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Enter values:

## Solution example

First Line Decomposition

To calculate the determinant of a matrix by first line decomposition, it is necessary to multiply each element of the given line by the corresponding minor;

The minors corresponding to a certain element are found by eliminating i-row , j-column from the matrix A, after which we find the determinant of the resulting matrix;
i, j are the numbers of the row and the column, in which the certain element is located;

After calculating the products of each element of the first row, in the corresponding minor, it is necessary to add and subtract them;
The sign of addition and subtraction changes in order, starting with the sign of addition;
Near the first product there is a plus sign, near the second a minus sign, etc.

det(A) =
 000 71 8 8 2 000 7 8 5 2 2 5 8 7 4 5 5 2
=

= a11 * A11 - a12 * A12 + a13 * A13 - a14 * A14;

So we find the minors of each element of the first row.

Find the minor element with index 11
To do this, it is necessary to exclude the 1 row and the 1 column from matrix A, after which we obtain the following matrix:

 000 8 5 2 000 5 8 7 5 5 2

Next, we calculate the determinant of this matrix.
It is equal to -57, this is the minor of element 11.

A11 =
 000 71 8 8 2 000 7 8 5 2 2 5 8 7 4 5 5 2
=
 000 8 5 2 000 5 8 7 5 5 2
= -57;

Find the minor element with index 12
To do this, it is necessary to exclude the 1 row and the 2 column from matrix A, after which we obtain the following matrix:

 000 7 5 2 000 2 8 7 4 5 2

Next, we calculate the determinant of this matrix.
It is equal to -57, this is the minor of element 12.

A12 =
 000 71 8 8 2 000 7 8 5 2 2 5 8 7 4 5 5 2
=
 000 7 5 2 000 2 8 7 4 5 2
= -57;

Find the minor element with index 13
To do this, it is necessary to exclude the 1 row and the 3 column from matrix A, after which we obtain the following matrix:

 000 7 8 2 000 2 5 7 4 5 2

Next, we calculate the determinant of this matrix.
It is equal to -3, this is the minor of element 13.

A13 =
 000 71 8 8 2 000 7 8 5 2 2 5 8 7 4 5 5 2
=
 000 7 8 2 000 2 5 7 4 5 2
= -3;

Find the minor element with index 14
To do this, it is necessary to exclude the 1 row and the 4 column from matrix A, after which we obtain the following matrix:

 000 7 8 5 000 2 5 8 4 5 5

Next, we calculate the determinant of this matrix.
It is equal to 21, this is the minor of element 14.

A14 =
 000 71 8 8 2 000 7 8 5 2 2 5 8 7 4 5 5 2
=
 000 7 8 5 000 2 5 8 4 5 5
= 21;

Now we need to calculate the product of the first element by its corresponding minor.
71 * (-57) = -4047;

Further, from this product, it is necessary to subtract the product of the second element by the corresponding minor.
-4047 - (8 * (-57)) = -4047 - (-456) = -3591;

Now to the result you need to add the product of the third element to the corresponding minor.
-3591 (8 * (-3)) = -3591 (-24) = -3615;

And finally, from the obtained result , it is necessary to subtract the product of the fourth element by the corresponding minor
-3615 - (2 * 21) = -3615 - 42 = -3657;

The result of this subtraction is the determinant of the matrix A

det(A) = (71 * (-57)) - (8 * (-57)) + (8 * (-3)) - (2 * 21) = -3657;

Sarrus

Let there be the following matrix A:

 000 2 5 6 000 5 8 2 3 5 7

To the right of matrix A, we add the first two columns;

 000 2 5 6 000 2 5 5 8 2 5 8 3 5 7 3 5
=

We take the products of the elements on the main diagonal and on the diagonals parallel to it with a plus sign;

= (a11a22a33) + (a12a23a31) + (a13a21a32) -

We take the products of the elements of the secondary diagonal and diagonals parallel to it with a minus sign;

= (a13a22a31) - (a11a23a32) - (a12a21a33) =

= (2 * 8 * 7) + (5 * 2 * 3) + (6 * 5 * 5) - (6 * 8 * 3) + (2 * 2 * 5) + (5 * 5 * 7) = -47;

Reduction to Triangular Form

We reduce the matrix to a triangular form, then the product of the elements of the main diagonal will give us the determinant;

det(A) =
 000 71 8 8 2 000 7 8 5 2 2 5 8 7 4 5 5 2
=

from 2 row we subtruct 1 row, multiplied by 0.09859;

from 3 row we subtruct 1 row, multiplied by 0.02817;

from 4 row we subtruct 1 row, multiplied by 0.05634;

=
 000 71 8 8 2 000 0 7.21128 4.21128 1.80282 0 4.77464 7.77464 6.94366 0 4.54928 4.54928 1.88732
=

from 3 row we subtruct 2 row, multiplied by 0.66211;

from 4 row we subtruct 2 row, multiplied by 0.63086;

=
 000 71 8 8 2 000 0 7.21128 4.21128 1.80282 0 0 4.98631 5.74999 0 0 1.89255 0.74999
=

from 4 row we subtruct 3 row, multiplied by 0.37955;

=
 000 71 8 8 2 000 0 7.21128 4.21128 1.80282 0 0 4.98631 5.74999 0 0 0 -1.43242
=

det(A) = 71 * 7.21128 * 4.98631 * -1.43242 = -3657;