I learnt recently recently that a linear transformation can be represented with the help of a matrix. However this matrix is not unique, it depends on the system of bases you take. But the determinant of all those matrices are same. So if I take a particular number, for example 21, and write down all possible matrices whose determinant is 21, will all these matrices represent the same transformation with different system of bases? Consider all endomorphisms.
[Math] Determinant of a transformation matrix
determinantlinear algebralinear-transformationsmatrices
Best Answer
No. if two matrices $A$ and $B$ represent the same linear transformation, then they are related by conjugation, $B=P^{-1}AP$, where $P$ is the change of basis. Such matrices are called similar. Two matrices may have same determinant but not be similar. They are similar if and only if they have the same canonical form.
For example
$$\left(\begin{matrix} 1 & 0\\ 0 & 1\end{matrix}\right)$$
and
$$\left(\begin{matrix} 1 & 1\\ 0 & 1\end{matrix}\right)$$
Both have determinant 1, but they are not similar.