Prove That $T(B)=AB-BA$ has $\dim \operatorname{Im}(T)\leq n^{2}-n$.
55230
Dimension of set of commutable matrices
50266
In the following figure prove that: $AK||BC$
69980
Proving that if $A$ is an $n\times n$ orthogonal matrix, then $\det(A) = \pm1$
41026
How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?
116389
If $\exp(t(A + B)) = \exp(tA) \exp(tB)$ for all $t \geq 0$ then $A,B$ commute
78224
Suppose $A$ is a general $n \times n$ matrix and $B$ is obtained by interchanging two rows of $A$. Prove that $\det(B) = -\det(A)$
78298
On the nilpotence of the matrix $AB-BA$
57771
Let $A\in M_{n\times n}(\Bbb R)$ so that $I\notin span(A,A^2,...,A^n)$. Prove that $\det(A)=0$.
101861
How do you prove that $tr(B^{T} A )$ is a inner product?
23095
Prove that T is a linear transformation
61103
Prove that $\mathbb{|Q| = |Q\times Q|}$
63036
Proving that $\dim(\mathrm{span}({I_n,A,A^2,...})) \leq n$
38850
How to show Von Neumann Trace inequality $ \text{Tr}(AB) \leq \sum_{i=1}^n \sigma_{A,i}\sigma_{B,i} $?
147827
How to calculate $|\operatorname {SL}_2(\mathbb Z/N\mathbb Z)|$?
6091
Similarity of products $AB$ and $BA$ of square matrices
47829
Prove that $\operatorname{trace}(A^TA) = 0\ \iff\ A = 0$.
158695
Prove that if $AB$ is invertible then $B$ is invertible.