I have three questions:
(i)Does rank of a square matrix same as the degree of its characterstic equation.
(ii)Do elementary row transformations of a given square matrix result in a characterstic equation different from that of given square matrix
(iii)Geometrically speaking what do cofactors of a square matrix mean. What is significance of some of them becoming zeros or all becoming zeros.
These are theoretical questions and so if you could refer to me some book or online resource which can explain these concepts it would be a great help.
Also it would be nice to know what other members think about these topics.
Thanks in advance
1) The degree of the characteristic equation for an $N\times N$ matrix is always $N$, regardless of the rank of the matrix. The lowest $m$ for which $x^m$ has a non-zero coefficient in the characteristic equation does have something to do with $n$ minus the rank.
2) Elementary row operations do change the characteristic equation. Similarity transformations ($P^{-1}MP$) do not.
3) There is no particular geometric significance to zeros in the cofactors, other than the fact that if the matrix is invertible, they correspond to zero entries in the inverse.