What is the eigenvalues of $T$ and what are their geometric multiplicity? $T: \Bbb R ^{n  \times n} \to \Bbb R ^{n  \times n}$ such that for every $A \in \Bbb R ^{n  \times n}$, $T(A)$ is the matrix $A$, but with the second row multiplied by 2.
What is the eigenvalues of $T$ and what are their geometric multiplicity? What is $\det(T)$?
I thought only about 2 eigenvalues:


*

*$2$ with geometric multiplicity $n$.

*$1$ with geometric multiplicity $n$.


Is that correct? Are there more?

You are correct that eigenvalue $\lambda = 2$ has multiplicity $n$, but the dimension of the $\lambda = 1$ eigenspace is equal to $n^2 - n$.  This linear transformation is diagonalizable.  Think about a set of eigenvectors with eigenvalue 1:  It is given by the set of matrices of all $0$'s except for a $1$ in the $i, j$ entry with i$\neq 2$.
The determinant gets a factor of $k$ for every entry multiplied by $k$:  In this case, that means $\det(T) = 2^n$.
