Elements of a square matrix 2+i 3 i-1 5/3 Calculation accuracy (signs after a comma)

 You entered the following elements of the massif 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 $AA^{-1}=A^{-1}A=E$

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)

$E=\begin{pmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{pmatrix}$

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.

$A=\begin{pmatrix}a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n}\\... & ... & ... & ...\\ a_{n1} & a_{n2} & ... & a_{nn}\end{pmatrix}$

has an inverse matrix

$A^{-1}=\frac{1}{Det(A)}\begin{pmatrix}A_{11} & A_{12} & ... & A_{1n}\\ A_{21} & A_{22} & ... & A_{2n}\\... & ... & ... & ...\\ A_{n1} & A_{n2} & ... & A_{nn}\end{pmatrix}$

Where A ij is the algebraic complement of the matrix

For example, the original matrix

$\begin{pmatrix} 1 & 3 & -5 & 11 \\ 1+i & 0 & -3 & 1.66 \\ 0.6 & 7 & 1 & -7 \\ -2 & -2-4i & 0 & 10 \end{pmatrix}$

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

$\begin{pmatrix}0.3232+0.3544i & -0.4955-0.7294i & 0.1294-0.4163i & -0.1827-0.5601i \\ 0.0327+0.018i & -0.0063-0.0194i & 0.1445+0.0318i & 0.0663+0.0057i \\ 0.025+0.2743i & -0.3066-0.4925i & 0.2052-0.1062i & 0.167-0.2943i \\ 0.064+0.0876i & -0.0926-0.1523i & 0.0421-0.0191i & 0.0745-0.0844i \\ \end{pmatrix}$

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 $\begin{pmatrix}1 & 3\\ 6 &5\end{pmatrix}$

The inverse matrix of the original is $\begin{pmatrix}-0.3846 & 0.2308 \\ 0.4615 & -0.0769 \\ \end{pmatrix}$

Matrix containing expressions

$\begin{pmatrix}1 & 3 & -i\\ 6 & sin(2+i) & 5\\ ln(\frac{3}{(4-i)}) & i^{1.433} & -8\end{pmatrix}$

after automatic conversion we get just such a matrix

$\begin{pmatrix}1 & 3 & 0-i \\ 6 & 1.4031-0.4891i & 5 \\ -0.318+0.245i & -0.6289+0.7775i & -8 \\ \end{pmatrix}$

And its inverse matrix has the following form

$\begin{pmatrix}-0.0595+0.0033i & 0.1828-0.005i & 0.1147+0.0043i \\ 0.3425-0.009i & -0.0609+0.0009i & -0.0392-0.0423i \\ -0.0238+0.032i & -0.0024-0.0002i & -0.1225+0.0029i \\ \end{pmatrix}$

Good calculations !!

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