Decomposition by:
1
Solution comments
Without description (answer only)
a
b
c
d
x
y
z
AC
i
Randomize
Number format
3131313131351515151515≈52188552198585858586
Round to
Digits after decimal point
10
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
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 1From
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
3
Iteration 2From
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
4
Iteration 3From
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
5
Matrix determinantdet(
A
) =
2
*
2
*
6
1
4
*
-1
2
25
=
-27
;
Answer
det(A)det(
A
) =
-27
;
SIZE4×4METHODTriangular Form(Gaussian elimination)