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Matrix Determinant Calculator

This calculator allows you to easily calculate the determinant of the matrix, as well as get a detailed solution. The calculator calculates the determinant for matrices with dimensions from 1 ร— 1 to 9 ร— 9.

An m ร— n matrix is a table of numbers with m rows and n columns. Matrix elements are denoted as aij, where i is the row number, j is the column number.
The determinant of a square matrix is a number that can be calculated from its elements.



Dimension of the matrix:
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Examples of calculating the determinant of a matrix


Calculate the determinant for a 1 ร— 1 matrix
det A =
7
= 7
For a 1 ร— 1 matrix, the determinant value is equal to the only element of this matrix
Go to calculator
Calculate the Determinant for a 2 ร— 2 Matrix
det A =
-25
0.453
= -8.25
det A = -2 โ‹… 3 - 0.45 โ‹… 5 = -8.25

For a 2 ร— 2 matrix, the determinant value is calculated by the formula
Go to calculator
Calculate the determinant for a 3 ร— 3 matrix
det A =
70.345-1
049
-231796
= 1453.585
det A = (7 โ‹… 4 โ‹… 96) - (7 โ‹… 9 โ‹… 17) - (0.345 โ‹… 0 โ‹… 96) + (0.345 โ‹… 9 โ‹… (-23)) + ((-1) โ‹… 0 โ‹… 17) - ((-1) โ‹… 4 โ‹… (-23)) = 1453.585

Solution

A =
a11 a12a13
a21 a22a23
a31a32a33
where,
a11 = 7
a12 = 0.345
a13 = -1
a21 = 0
a22 = 4
a23 = 9
a31 = -23
a32 = 17
a33 = 96
For a 3 ร— 3 matrix, the determinant value is calculated by the formula

det A =
a11a12a13
a21 a22a23
a31 a32a33
= a11
a22a23
a32a33
- a12
a21a23
a31a33
+ a13
a21a22
a31a32
= a11a22 a33 - a11a23a32 - a12a21a33 + a12a23a31 + a13a21a32 - a13a22a31

Then, substituting the values of the matrix elements, we get
det A = (7 โ‹… 4 โ‹… 96) - (7 โ‹… 9 โ‹… 17) - (0.345 โ‹… 0 โ‹… 96) + (0.345 โ‹… 9 โ‹… (-23)) + ((-1) โ‹… 0 โ‹… 17) - ((-1) โ‹… 4 โ‹… (-23)) = 1453.585
Go to calculator
Calculate the Determinant for a 4 ร— 4 Matrix
det A =
-3120 4
31572
00-0.4534.67
162713
= 356897.28
det A = (-1)1 โ‹… (-3) โ‹… (-3248.16) + (-1)2 โ‹… 12 โ‹… (-28956.55) + (-1)3 โ‹… 0 โ‹… (-6275.27) + (-1)4 โ‹… 4 โ‹… 81.45 = 356897.28

Solution

A =
a11a12a13 a14
a21a22 a23a24
a31 a32a33a34
a41a42a43 a44
where,
a11 = -3
a12 = 12
a13 = 0
a14 = 4
a21 = 31
a22 = 5
a 23 = 7
a24 = 2
a31 = 0
a32 = 0
a33 = -0.45
a34 = 34.67
a41 = 1
a42 = 6
a43 = 27
a44 = 13
For an n ร— n matrix, the determinant value is calculated by the formula

M1j - additional minor to the element a1j, calculated from the initial matrix A by deleting the first row and the jth column

The value of n = 4, so it is necessary to calculate 4 additional minors by deleting the first row and j-th column

M11 =
-31204
31572
00-0.4534.67
162713
=
572
0-0.4534.67
6 2713
= -3248.16

M12 =
-31204
31572
00-0.4534.67
162713
=
3172
0-0.4534.67
12713
= -28956.55

M13 =
-3 1204
31572
00-0.4534.67
162713
=
3152
0034.67
1613
= -6275.27

M14 =
-3120 4
31572
00-0.4534.67
162713
=
31 57
00-0.45
1627
= 81.45


Based on the above formula, we will write down the amount

det A = (-1)1 โ‹… a11 โ‹… M11 + (-1)2 โ‹… a12 โ‹… M12 + (-1)3 โ‹… a13 โ‹… M13 + (-1)4 โ‹… a14 โ‹… M14

det A = (-1)1 โ‹… (-3) โ‹… det
572
0-0.4534.67
627 13
+ (-1)2 โ‹… 12 โ‹… det
3172
0-0.4534.67
1 2713
+ (-1)3 โ‹… 0 โ‹… det
3152
0034.67
1613
+ (-1)4 โ‹… 4 โ‹… det
3157
00-0.45
1627
= (-1)1 โ‹… (-3) โ‹… (-3248.16) + (-1)2 โ‹… 12 โ‹… (-28956.55) + (-1)3 โ‹… 0 โ‹… (-6275.27) + (-1)4 โ‹… 4 โ‹… 81.45 = 356897.28
Go to calculator