Inverse Matrix

1. Enter numbers in "Matrix".
2. ※You can add "Rows" and "Columns".
3. ※You can only calculate the inverse matrix if it is a square matrix.
4. The "inverse matrix" is calculated automatically.

What is an inverse matrix?

An inverse matrix is ​​a concept of linear algebra in mathematics, and is a square matrix (a matrix with the same number of rows and columns) that satisfies certain conditions. The inverse matrix is ​​defined as follows.

Definition of an inverse matrix

For a square matrix A, if matrix B exists and satisfies the following relationship, matrix B is called the inverse matrix of matrix A.

A⋅B=B⋅A=I

Here, I is the identity matrix (a matrix with all diagonal elements equal to 1 and all other elements equal to 0).

The inverse of matrix A is usually written as A₋₁.

Characteristics of an inverse matrix

Conditions for the existence of an inverse matrix

An inverse matrix exists only if matrix A is a regular matrix (a matrix whose determinant is not 0).

An inverse matrix does not exist for a non-regular matrix (a matrix whose determinant is 0).

Properties of an inverse matrix

1. (A^-1)^-1 = A: The inverse of an inverse matrix is ​​the original matrix.

2. (A · B)^-1 = B^-1 · A^-1: The inverse of a matrix product is the product of the inverse matrices of each matrix, with the order reversed.

3. (A^T)^-1 = (A^-1)^T: The inverse of a transposed matrix is ​​the transpose of its inverse.

4. A^-1 · A = A · A^-1 = I: The product of a matrix and its inverse is the identity matrix.

Notes

When calculating an inverse matrix, the following points must be taken into consideration.

The matrix must be a square matrix.

The determinant must not be 0 (it must be a regular matrix).

Matrices with higher dimensions are more complex to calculate, so numerical methods and software are commonly used.

Inverse matrices play an important role in mathematics and science and technology and have many applications.

To calculate the inverse of a matrix, use the following steps:

1. Determine the matrix

First, for an inverse to exist, the original matrix A must be a square matrix.

2. Calculating the determinant

Next, calculate the determinant. Let the determinant of matrix A be det(A).

Example: 2x2 matrix

A = [ a₁₁ a₁₂ ]
      [ a₂₁ a₂₂ ]

To calculate the determinant:

det(A) = a₁₁ · a₂₂ - a₁₂ · a₂₁

If det(A) = 0 then there is no inverse matrix.

3. Calculating the cofactor matrix

Calculate the cofactor for each element of the matrix A.

What is a cofactor:

It is the signed determinant of the minor matrix (minor) excluding the rows and columns containing certain elements of the matrix.

4. Calculating the adjoint matrix (transposed matrix)

Create a matrix by placing the calculated cofactors, and then transpose it to find the adjoint matrix.

5. Calculating the inverse matrix

Finally, the inverse matrix A^-1 is calculated as follows:

A^-1 = (1 / det(A)) · adj(A)

Where adj(A) is the adjoint matrix.

Example: Inverse of a 2x2 matrix

Matrix A:

A = [ 2 3 ]
       [ 1 4 ]

1. Calculate the determinant:

det(A) = 2 · 4 - 3 · 1 = 8 - 3 = 5

2. Calculate the cofactor matrix:

adj(A) = [ 4 -3 ]
          [-1 2 ]

3. Find the inverse matrix:

A^-1 = (1 / 5) · adj(A)

A^-1 = [ 4/5 -3/5 ]
            [-1/5 2/5 ]