The original polynomial f(x) (its coefficients) The argument is a square matrix with elements
 Polynomial Variable x= Calculation result

We will consider in this material one of the time-consuming tasks in higher mathematics, which sounds like this: Find what the polynomial is given

$f(x)=a_0x^{n}+a_1x^{n-1}+a_2x^{n-2}+.....+a_{n-1}x+a_n$

if the argument is a square matrix, then there is $x=\begin {pmatrix}x_{00}&x_{10}&...&x_{i0}\\x_{01}&x_{11}&...&x_{i1}\\ ... & ...&...&... \\ x_0j} & x_{1j}&...&x_{ij}\end{pmatrix}$

And if the calculation principle itself is clear, especially if you have perfectly understood how to multiply matrices, then direct calculation, for me personally, is considered routine, which should be avoided if possible.

I would like to say right away where this calculator will come in handy. For teachers, teachers, for textbook creators, for those who need to create original tasks on this topic.

It is also useful for students or postgraduates who write essays, term papers, diplomas.

For everyone else, this is an easy way to check the error in a given example, to solve, without long intermediate calculations, the task.

When the calculator was written, it turned out that the sites that were dedicated to this topic contained errors in intermediate calculations and as a result were incorrect.

This calculator, I hope, is free from errors and you will be able to safely solve any examples.

Like the vast majority of calculators on this site, the values of both the coefficients of the polynomial and the elements of the matrix can be complex values.

Such a thing at the end of 2017, you will not find anywhere else, except of course for special created mathematical programs.

Shall we proceed to the examples?

Find the value of the polynomial $f (x)=2x^2-3x+4$ from matrix $x=\begin {pmatrix}-1&2&0\\2&1&-3\\{0}&-1&2\end{pmatrix}$

 Polynomial $f (x) = (2)* x^{2}+(-3)*x+(4)$ Variable x= $x = \begin{pmatrix}-1 & 2 & 0 \\ 2 & 1 & -3 \\ 0 & -1 & 2 \\ \end{pmatrix}$ Calculation result $f(x) = \begin{pmatrix}17 & -6 & -12 \\ -6 & 17 & -9 \\ -4 & -3 & 12 \\ \end{pmatrix}$

Another example

What is the polynomial $f (x)=ix^5+(2-i)x^2-11x$ if $x=\begin {pmatrix}2 &2-i&3\\-7&0&i\\1+i&2&0\end{pmatrix}$

 Polynomial $f (x) = (i)*x^{5}+(2-i)*x^{2}+(-11)*x$ Variable x= $x = \begin{pmatrix}2 & 2-i & 3 \\ -7 & 0 & i \\ 1+i & 2 & 0 \\ \end{pmatrix}$ Calculation result $f(x) = \begin{pmatrix}2774-2058i & -1092-741i & -66-1293i \\ -1336+2039i & 1937+1391i & 995+2236i \\ 1300+279i & 389+117i & 543+401i \\ \end{pmatrix}$

Find the value of the polynomial $f(x)=x^4-x-1$ from the complex matrix

$x=\begin{pmatrix}-1 & 0 & 0 & -1 & 0 & 0 \\ 0 & -1 & 1 & 0 & -1 & 1 \\ -1 & 0 & i & i & i & -1 \\ -1 & -1 & i & -1 & -1 & i \\ 0 & 1 & -1 & 0 & -1 & i \\ 1 & 0 & 0 & 0 & i & 0 \\ \end{pmatrix}$

 Polynomial $f (x) = x^{4}+(-1)*x+(-1)$ Variable x= $x = \begin{pmatrix}-1 & 0 & 0 & -1 & 0 & 0 \\ 0 & -1 & 1 & 0 & -1 & 1 \\ -1 & 0 & i & i & i & -1 \\ -1 & -1 & i & -1 & -1 & i \\ 0 & 1 & -1 & 0 & -1 & i \\ 1 & 0 & 0 & 0 & i & 0 \\ \end{pmatrix}$ Calculation result $f(x) = \begin{pmatrix}7+2i & 3 & -2-2i & 6+i & 1+2i & -3-7i \\ 4+7i & -3-5i & 6+i & 5 & 3-2i & 11 \\ 3-5i & 3+i & -7+i & -2-2i & -4+9i & -4+i \\ 7+6i & 5-i & -1 & 5+4i & 0+6i & 1-2i \\ -2-5i & 3+4i & -8-5i & -2-i & -9-9i & -8+i \\ -5-i & -2+i & -2+3i & -5 & -8+4i & 0+2i \\ \end{pmatrix}$

Successful calculations !

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