Matrix determinant calculator

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3131313131351515151515≈83137
2
2510
=Solve

  How to find the determinant via triangular form

Apply elementary row operations to reduce the matrix to upper triangular form, tracking each row swap (sign change), row scaling (multiplicative factor), and elimination. The determinant equals the product of the diagonal entries, adjusted by the tracked factors.

  Triangular form worked example (4×4)

Write the initial matrix
A
:
A
=
2
4
1
0
-1
0
3
2
3
1
-1
1
1
-2
4
5
To find the determinant of matrix
A
need to do the following:
1)
The determinant of a triangular matrix equal the product of the elements of the main diagonal;
2)
To find the determinant of matrix A, need to reduce it to a triangular form and then multiply the elements of the main diagonal;
3)
To reduce the matrix A to a triangular form, use the Gaussian elimination;
det(
A
) =
a
0
1,1
·
a
0
2,2
· ··· ·
a
0
n,n
// where
a
a is an element of matrix A;
det(
A
) =
2
4
1
0
-1
0
3
2
3
1
-1
1
1
-2
4
5
=
2
Iteration 1
From
2
th row we subtract
1
th row, multiplied by
2
;
From
3
th row we subtract
1
th row, multiplied by
1
2
;
2
0
0
0
-1
2
3
1
2
2
3
-5
-2
1
2
1
1
-4
3
1
2
5
a
0
2,1
=
4
- (
2
*
2
)
=
0
;
a
0
2,2
=
0
- (
2
*
-1
)
=
2
;
a
0
2,3
=
1
- (
2
*
3
)
=
-5
;
a
0
2,4
=
-2
- (
2
*
1
)
=
-4
;
a
0
3,1
=
1
- (
1
2
*
2
)
=
0
;
a
0
3,2
=
3
- (
1
2
*
-1
)
=
3
1
2
;
a
0
3,3
=
-1
- (
1
2
*
3
)
=
-2
1
2
;
a
0
3,4
=
4
- (
1
2
*
1
)
=
3
1
2
;
Hide description
3
Iteration 2
From
3
th row we subtract
2
th row, multiplied by
1
3
4
;
From
4
th row we subtract
2
th row, multiplied by
1
;
2
0
0
0
-1
2
0
0
3
-5
6
1
4
6
1
-4
10
1
2
9
a
0
3,2
=
3
1
2
- (
1
3
4
*
2
)
=
0
;
a
0
3,3
=
-2
1
2
- (
1
3
4
*
-5
)
=
6
1
4
;
a
0
3,4
=
3
1
2
- (
1
3
4
*
-4
)
=
10
1
2
;
a
0
4,2
=
2
- (
1
*
2
)
=
0
;
a
0
4,3
=
1
- (
1
*
-5
)
=
6
;
a
0
4,4
=
5
- (
1
*
-4
)
=
9
;
Hide description
4
Iteration 3
From
4
th row we subtract
3
th row, multiplied by
24
25
;
2
0
0
0
-1
2
0
0
3
-5
6
1
4
0
1
-4
10
1
2
-1
2
25
a
0
4,3
=
6
- (
24
25
*
6
1
4
)
=
0
;
a
0
4,4
=
9
- (
24
25
*
10
1
2
)
=
-1
2
25
;
Hide description
5
Matrix determinant
det(
A
) =
2
*
2
*
6
1
4
*
-1
2
25
=
-27
;
Answer
det(A)
det(
A
) =
-27
;
SIZE4×4METHODTriangular Form(Gaussian elimination)

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