Online Calculator: Matrix Determinant

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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;

Answer:det(A) = -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;

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;

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;

Answer:det(A) = -3657