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: and
One solution is the matrix method.
So our resulting vector has values
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
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
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.
Naturally, all this works in the field of complex numbers.
That is, if we have a matrix
Then the resulting vector has the form
The restriction, again, is one  the matrix is not more than 10 by 10.
I hope this helps someone in their work.