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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 =
 2 0 4 7
A-1 =
 1โ2 0 2-โ 7 1โ7
A-1 =
 0.5 0 -0.285714 0.142857

Solution

Let us calculate the inverse of the matrix A using the matrix 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 =
 2 0 4 7
=
 7
= 7

M12 =
 2 0 4 7
=
 4
= 4

M21 =
 2 0 4 7
=
 0
= 0

M22 =
 2 0 4 7
=
 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
 A11 A12 A21 A22
=
 7 -4 0 2

3. Transpose the attached matrix adj(A)
 7 0 -4 2
4. We divide all elements of the matrix adj(A)T by the determinant of the initial matrix A
A-1 =
 7โโ14 0โโ14 -4โโ14 2โโ14
A-1 =
 1โ2 0 2-โ 7 1โ7
A-1 =
 0.5 0 -0.285714 0.142857
Go to calculator
Calculate the inverse of a 3 ร 3 matrix A
A =
 4 1 8 -5 0 12 73 94 0
A-1 =
 282โโโโ1849 188-โโโโ 1849 3-โโโโ 1849 219-โโโโ 1849 146โโโโ1849 22โโโโ1849 235โโโโ3698 303โโโโ7396 5-โโโโ 7396
A-1 =
 0.152515 -0.101677 -0.0016225 -0.118442 0.0789616 0.0118983 0.0635479 0.0409681 -0.000676041

Solution

Let us calculate the inverse of the matrix A using the matrix 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 =
 4 1 8 -5 0 12 73 94 0
=
 0 12 94 0
= -1128

M12 =
 4 1 8 -5 0 12 73 94 0
=
 -5 12 73 0
= -876

M13 =
 4 1 8 -5 0 12 73 94 0
=
 -5 0 73 94
= -470

M21 =
 4 1 8 -5 0 12 73 94 0
=
 1 8 94 0
= -752

M22 =
 4 1 8 -5 0 12 73 94 0
=
 4 8 73 0
= -584

M23 =
 4 1 8 -5 0 12 73 94 0
=
 4 1 73 94
= 303

M31 =
 4 1 8 -5 0 12 73 94 0
=
 1 8 0 12
= 12

M32 =
 4 1 8 -5 0 12 73 94 0
=
 4 8 -5 12
= 88

M33 =
 4 1 8 -5 0 12 73 94 0
=
 4 1 -5 0
= 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
 A11 A12 A13 A21 A22 A23 A31 A32 A33
=
 -1128 876 -470 752 -584 -303 12 -88 5

3. Transpose the attached matrix adj(A)