Prove that $ AB = BA $ if and only if $ \det\begin{bmatrix} b & a - c \\ e & d - f \\ \end{bmatrix} = 0 $
33560
Show that matrices $A$ and $B$ commute if and only if $\begin{vmatrix} b & a-c \\ e & d-f\end{vmatrix}=0.$
69980
Proving that if $A$ is an $n\times n$ orthogonal matrix, then $\det(A) = \pm1$
128760
If $A$ and $B$ two $n\times n$ matrix such that $AB=BA$, then $\dim(A\cap B)$?
31783
Is this proof of $ab = 0$ correct?
105921
Why do we have $ab=ba$?
142640
Prove determinant of a matrix $=\prod_{j<i}(a_i-a_j)$
7712
How to prove the both identity (matrix)
63119
Let $M=\big( \begin{smallmatrix} A & B \\ C & D \end{smallmatrix} \big)$. Prove $\det(M)=\det(A)\cdot \det(D-C·A^{-1}·B)$.
81181
Prove that $\det(M-I)=0$ if $\det(M)=1$ and $MM^T=I$
123555
Proving that if $ab=e$ then $ba=e$
51800
We have matrix $A\in M_{n-1\times n}(\mathbb Z)$ so that the sum of entries in each row is zero. Prove that $\det(AA^T)=nk^2.$
125348
Does $\det(A + B) = \det(A) + \det(B)$ hold?
140290
Solving $AB - BA = C$
85167
When A and B are of different order given the $\det(AB)$,then calculate $\det(BA)$
60900
If $A^2=-I$, Prove that $\det{A}=1$
33891
If $\mathbf A$ and $\mathbf B$ are Hermitian, when does $\det(\mathbf A-\lambda\mathbf B)=0$ have only real roots?
110484
Condition on a matrix so that its determinant is $2$