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Inverse Matrix Calculator

This calculator allows you to easily calculate the inverse of a matrix using a matrix algebraic additions, as well as get a detailed solution. The calculator calculates the inverse matrix for matrices with dimensions from 2 ร— 2 to 9 ร— 9. With this calculator, you can quickly learn how to calculate the inverse matrix, thanks to the detailed solution algorithm.


An m ร— n matrix is โ€‹โ€‹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.
A matrix A-1 is called inverse to matrix A if A โ‹… A-1 = A-1 โ‹… A = E, where E is the identity matrix.
To calculate the inverse matrix, the matrix A must be square and not equal to zero.



Dimension of the matrix:
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Examples of calculating the inverse matrix

Calculate the inverse of a 2 ร— 2 matrix A
A =
20
4 7
A-1 =
1
โ€”
2
0
2
-โ€”
7
1
โ€”
7
A-1 =
0.5 0
-0.2857142857142860.142857142857143

Solution

Let us calculate the inverse of the matrix A using the matrix of algebraic complements.
adj(A) - adjoint matrix made up of algebraic complements
det(A) - determinant of matrix A

1. Calculate the determinant for the matrix A
det A = 14
(If you want to get a detailed solution for calculating the determinant, then use the matrix determinant calculator )

2. Calculate the adjoint matrix adj (A) composed of algebraic complements. To do this, replace each element of the initial matrix aij on its algebraic complement Aij
Mij - additional minor calculated from the initial matrix A by deleting the i-th row and j-th column
The initial matrix A consists of 4 elements, therefore we need to calculate 4 additional minors Mij

M11 =
20
47
=
7
= 7

M12 =
20
47
=
4
= 4

M21 =
20
47
=
0
= 0

M22 =
20
47
=
2
= 2

Now we write the value of all elements of the adjoined matrix adj(A)

A11 = (-1)1 + 1 โ‹… M11 = (-1)2 โ‹… 7 = 7
A12 = (-1)1 + 2 โ‹… M12 = (-1)3 โ‹… 4 = -4
A21 = (-1)2 + 1 โ‹… M21 = (-1)3 โ‹… 0 = 0
A22 = (-1)2 + 2 โ‹… M22 = (-1) 4 โ‹… 2 = 2
adj(A) =
A11A12
A21A22
=
7-4
02

3. Transpose the attached matrix adj(A)
adj(A)T =
70
-4 2
4. We divide all elements of the matrix adj(A)T by the determinant of the initial matrix A
A-1 =
7
โ€”โ€”
14
0
โ€”โ€”
14
-4
โ€”โ€”
14
2
โ€”โ€”
14
A-1 =
1
โ€”
2
0
2
-โ€”
7
1
โ€”
7
A-1 =
0.50
-0.2857142857142860.142857142857143
Go to calculator
Calculate the inverse of a 3 ร— 3 matrix A
A =
418
-5012
73940
A-1 =
282
โ€”โ€”โ€”โ€”
1849
188
-โ€”โ€”โ€”โ€”
1849
3
-โ€”โ€”โ€”โ€”
1849
219
-โ€”โ€”โ€”โ€”
1849
146
โ€”โ€”โ€”โ€”
1849
22
โ€”โ€”โ€”โ€”
1849
235
โ€”โ€”โ€”โ€”
3698
303
โ€”โ€”โ€”โ€”
7396
5
-โ€”โ€”โ€”โ€”
7396
A-1 =
0.152514872904273-0.101676581936182-0.00162249864791779
-0.1184424012979990.0789616008653326 0.0118983234180638
0.06354786371011360.0409680908599243 -0.000676041103299081

Solution

Let us calculate the inverse of the matrix A using the matrix of algebraic complements.
adj(A) - adjoint matrix made up of algebraic complements
det(A) - determinant of matrix A

1. Calculate the determinant for the matrix A
det A = -7396
(If you want to get a detailed solution for calculating the determinant, then use the matrix determinant calculator )

2. Calculate the adjoint matrix adj (A) composed of algebraic complements. To do this, replace each element of the initial matrix aij on its algebraic complement Aij
Mij - additional minor calculated from the initial matrix A by deleting the i-th row and j-th column
The initial matrix A consists of 9 elements, therefore we need to calculate 9 additional minors Mij

M11 =
418
-5012
73940
=
012
940
= -1128

M12 =
418
-50 12
73940
=
-512
730
= -876

M13 =
418
-5012
73940
=
-50
7394
= -470

M21 =
418
-5012
73 94 0
=
18
940
= -752

M22 =
418
-50 12
73 940
=
48
730
= -584

M23 =
418
-5012
73940
=
41
7394
= 303

M31 =
4 18
-5 012
73 940
=
18
012
= 12

M32 =
418
-5012
73940
=
48
-512
= 88

M33 =
418
-5012
7394 0
=
41
-50
= 5

Now we write the value of all elements of the adjoined matrix adj(A)

A11 = (-1)1 + 1 โ‹… M11 = (-1)2 โ‹… (-1128) = -1128
A12 = (-1)1 + 2 โ‹… M12 = (-1)3 โ‹… (-876) = 876
A13 = (-1)1 + 3 โ‹… M13 = (-1)4 โ‹… (-470) = -470
A21 = (-1)2 + 1 โ‹… M21 = (-1)3 โ‹… (-752) = 752
A22 = (-1)2 + 2 โ‹… M22 = (-1)4 โ‹… (-584) = -584
A23 = (-1)2 + 3 โ‹… M23 = (-1)5 โ‹… 303 = -303
A31 = (-1)3 + 1 โ‹… M31 = (-1)4 โ‹… 12 = 12
A32 = (-1)3 + 2 โ‹… M32 = (-1)5 โ‹… 88 = -88
A33 = (-1)3 + 3 โ‹… M33 = (-1)6 โ‹… 5 = 5
adj(A) =
A11A12A13
A21A22A23
A31 A32A33
=
-1128876-470
752-584-303
12-885

3. Transpose the attached matrix adj(A)
adj(A)T =
-112875212
876-584-88
-470-3035
4. We divide all elements of the matrix adj(A)T by the determinant of the initial matrix A
A-1 =
-1128
โ€”โ€”โ€”โ€”โ€”
-7396
752
โ€”โ€”โ€”โ€”โ€”
-7396
12
โ€”โ€”โ€”โ€”โ€”
-7396
876
โ€”โ€”โ€”โ€”โ€”
-7396
-584
โ€”โ€”โ€”โ€”โ€”
-7396
-88
โ€”โ€”โ€”โ€”โ€”
-7396
-470
โ€”โ€”โ€”โ€”โ€”
-7396
-303
โ€”โ€”โ€”โ€”โ€”
-7396
5
โ€”โ€”โ€”โ€”โ€”
-7396
A-1 =
282
โ€”โ€”โ€”โ€”
1849
188
-โ€”โ€”โ€”โ€”
1849
3
-โ€”โ€”โ€”โ€”
1849
219
-โ€”โ€”โ€”โ€”
1849
146
โ€”โ€”โ€”โ€”
1849
22
โ€”โ€”โ€”โ€”
1849
235
โ€”โ€”โ€”โ€”
3698
303
โ€”โ€”โ€”โ€”
7396
5
-โ€”โ€”โ€”โ€”
7396
A-1 =
0.152514872904273-0.101676581936182 -0.00162249864791779
-0.1184424012979990.0789616008653326 0.0118983234180638
0.06354786371011360.0409680908599243 -0.000676041103299081
Go to calculator