Complex inverse matrix online

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Inverse square matrix

In this material online the return matrix calculates from square set. Works with complex numbers.

The matrix is called return for a square matrix of A if 

where E - a single matrix (i.e. a matrix on the main diagonal of which there are units, and all other elements are equal to zero) 

A square matrix A is called degenerate if its determinant is zero, and non-degenerate otherwise .

If the matrix A has the inverse, then this matrix is ​​non-degenerate.

The converse is also true. Every non-degenerate matrix.

has an inverse matrix

Where A _ij is the algebraic complement of the matrix

For example, the original matrix

And this is the opposite, with rounding to 4 decimal places

What is the practical value of the inverse matrix? Where can we use it?

The simplest and most illustrative example.

We have a system of equations

We need to express  and  through  and 

if we take from the matrix

 the opposite, then we get 

And therefore our decision looks like this

Using an inverse matrix, for example, we need to create diophantine equations from a common matrix.

Some more examples

Source matrix 

The inverse matrix of the original is 

Matrix containing expressions

after automatic conversion we get just such a matrix

And its inverse matrix has the following form

Good calculations !!