Elements which have to be in a matrix
 Set a vector and the resulting vector

A fairly simple task that occurs in school textbooks: Find the vector product of two vectors

For example:  $a(3,4,-4)$ and $b(1,-2,1)$

One solution is the matrix method.

$\begin{pmatrix}i & j & k \\ 3 & 4 & -4 \\ 1 & -2 & 1 \\ \end{pmatrix}=( -4 )i - ( 7 )j + ( -10 )k$

So our resulting vector has values

$ab=(-4,-7,-10)$

All calculations are performed in the right coordinate system. If you need to multiply the vectors in the left coordinate system, then each resulting value must be taken with the opposite sign.

In the left coordinate system, our answer will be $ab=(4,7,10)$

## Extending the original theme

Consider the more general problem of how to calculate the "resulting vector" when there is a matrix without one top row. Rather, each element of the top row is an unknown variable.

When we have such a matrix

$\begin{pmatrix}i & j & k & l & m & n \\ 5 & 2 & 5 & 5 & 4 & 6 \\ 1 & 4 & 5 & 3 & 5 & 2 \\ 5 & 4 & 2 & 6 & 5 & 1 \\ 4 & 6 & 5 & 2 & 5 & 4 \\ 6 & 6 & 6 & 2 & 6 & 1 \\ \end{pmatrix}$

And you need to decompose it into a "vector"

Practical application is very important, to me, it helps to solve systems of linear Diophantine equations

In addition, this calculator easily checks the correctness of the solution of systems of equations.

The final solution to the given matrix will be an expression.

$( -76 )i + ( 1127 )j + ( 1108 )k +\\\\+ ( 1046 )l + ( -2426 )m + ( -490 )n$

Naturally, all this works in the field of complex numbers.

That is, if we have a matrix

$\begin{pmatrix}i &j &k& l\\1 &2 &i &5\\ 0 &-i &1 &4 \\-i &3 &1+i &2 \end{pmatrix}$

Then the resulting vector has the form

$( -16-1i )i + ( -2-1i )j + ( -7-10i )k +\\\\+ ( 2+2i )l$

The restriction, again, is one - the matrix is ​​not more than 10 by 10.

I hope this helps someone in their work.

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