Show that if $A \in M_{nxn}(k)$ and $det A \neq 0$ then for each matrix $B \in M_{nxn}(k)$: $AB$ and $BA$ are similar
66709
Are the matrix products $AB$ and $BA$ similar?
128990
Prove that, at least one of the matrices $A+B$ and $A-B$ has to be singular
106986
Eigenvalues of $AB$ and $BA$
25488
Let $A,B$ be $m\times m$ matrices such that $AB$ is invertible. Show $A,B$ invertible.
70606
Show that the LU decomposition of matrices of the form $\left[\begin{smallmatrix}0& x\\0 & y\end{smallmatrix}\right]$ is not unique
141173
Are there matrices such that $AB=I$ and $BA \neq I$
33591
Product of two matrices sharing the same eigenvector
57771
Let $A\in M_{n\times n}(\Bbb R)$ so that $I\notin span(A,A^2,...,A^n)$. Prove that $\det(A)=0$.
151443
Find two $2 \times 2$ matrices $A$ and $B$ with the same rank, determinant,...but they are not similar.
2798
A and B are nxn matrices, $A =B^TB$. Prove that if rank(B)=n, A is pos def, and if rank(B)<b, A is pos semi-def.
71040
If $ \operatorname{Tr}(M^k) = \operatorname{Tr}(N^k)\;\forall 1\leq k \leq n$, then how do we show the $M$ and $N$ have the same eigenvalues?
9574
How to show that $\det(AB) =\det(A) \det(B)$?
32487
Proving $\det \left( \begin{smallmatrix} A & -B \\ B & A \end{smallmatrix} \right) =|\det(A+iB)|^2$
69641
Determinant identity: $\det M \det N = \det M_{ii} \det M_{jj} - \det M_{ij}\det M_{ji}$
114250
If $V_0$ is the subspace of matrices of the form $C=AB-BA$ for some $A,B$ in a vector space $V$ then $V_0=\{A\in V|\operatorname{Trace} (A)=0\}$
152932
Proof: $\det\pmatrix{\langle v_i , v_j \rangle}\neq0$ $\iff \{v_1,\dots,v_n\}~\text{l.i.}$
147903
Similar matrices-Identity