Question

what is a matrix ? an open question i mean , please !

Answer 1

An a,b,c,d matrix.
\[\left[\begin{matrix}a & b \\ c & d\end{matrix}\right]\]
It is a 2 x 2 matrix as it has 2 rows and 2 collums.
A Matrix is a mathematical object, It's usually used whem operating with vectors.
e.g. \[\left[\begin{matrix}a & b \\ c & d\end{matrix}\right]\times \left(\begin{matrix}x \\ y\end{matrix}\right)=\left(\begin{matrix}ax+by \\ cx+dy\end{matrix}\right)\]
In this case the (x,y) vector may represent a place in a 2 dimensional room. Then the the matrix is a function operating on (x,y) giving a new vector.
A matrix scales and rotates the room.
A scaling Matrix is
\[\left[\begin{matrix}m & 0 \\ 0 & n\end{matrix}\right]\]
because then (x,y) turns to (nx, my). The x-dimension is scaled by n. The y-dimension is scaled by m.
A rotation matrix is
\[\left[\begin{matrix}\cos(\alpha) & -\sin(\alpha) \\ \sin(\alpha) & \cos(\alpha)\end{matrix}\right]\]
It rotates (x,y) by alpha.
A combination of both would be
\[\left[\begin{matrix}n & 0 \\ 0 & m\end{matrix}\right]\times \left[\begin{matrix}\cos(\alpha) & -\sin(\alpha) \\ \sin(\alpha) & \cos(\alpha)\end{matrix}\right]=\left[\begin{matrix}n \times \cos(\alpha) & -n \times \sin(\alpha) \\ m \times \sin(\alpha) & m \times \cos(\alpha)\end{matrix}\right]\]
A combination of scaling matrices is a scaling matrix. A combination of rotation matrices is a rotation matrix.
This also works with more dimensions. Just try it out.