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.
= 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:
Next, we calculate the determinant of this matrix.
It is equal to -57, this is the minor of element 11.
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:
Next, we calculate the determinant of this matrix.
It is equal to -57, this is the minor of element 12.
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:
Next, we calculate the determinant of this matrix.
It is equal to -3, this is the minor of element 13.
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:
Next, we calculate the determinant of this matrix.
It is equal to 21, this is the minor of element 14.
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;
Answer:det(A) = -3657
Sarrus
Let there be the following matrix A: To the right of matrix A, we add the first two columns; 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;
Answer:det(A) = -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; 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;
Answer:det(A) = -3657