Elements of a square matrix
Calculation accuracy (signs after a comma)


You entered the following elements of the massif
The set matrix
Inverse square matrix
Return matrix 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 inverse matrix

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) 

единичная матрица

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.

nondegenerate matrix

has an inverse matrix

inverse matrix

Where A ij is the algebraic complement of the matrix

For example, the original matrix

inverse matrix - calculation

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

inverse complex matrix

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

5x_1 + 3x_2 = h_1 \\ 3x_1 + 2x_2 = h_2

We need to express x_1 and x_2 through h_1 and h_2

if we take from the matrix

\ begin {pmatrix} 5 & 3 \\ 3 & 2 \ end {pmatrix} the opposite, then we get \ begin {pmatrix} 2 & -3 \\ -3 & 5 \ end {pmatrix}

And therefore our decision looks like this

x_1 = 2h_1-3h_2 \\ x_2 = -3h_1 + 5h_2

 

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

 

Some more examples

Source matrix origin matrix

The inverse matrix of the original is inventory matrix


Matrix containing expressions

complex matrix

 

after automatic conversion we get just such a matrix

complex matrix example

 

And its inverse matrix has the following form

inverse complex matrix

Good calculations !!

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