query: If $B$ is nilpotent and $AB=BA$ then $\det(A+B)=\det(A)$ (Asking for other method)
positive_passages.docid: 142980
positive_passages.text: If $\,A^k=0$ and $AB=BA$, then $\,\det(A+B)=\det B$
negative_passages.docid: 7034
negative_passages.text: Prove that $A+I$ is invertible if $A$ is nilpotent
negative_passages.docid: 14873
negative_passages.text: $1+a$ and $1-a$ in a ring are invertible if $a$ is nilpotent
negative_passages.docid: 10348
negative_passages.text: Prove that if the $\gcd(a,b)=1$, then $\gcd(a+b,ab)=1$
negative_passages.docid: 17032
negative_passages.text: If $ab=ba$, Prove $a^2$ commutes with $b^2$
negative_passages.docid: 125030
negative_passages.text: Prove that $\det(A)=p_1p_2-ba={bf(a)-af(b)\over b-a}$
negative_passages.docid: 56097
negative_passages.text: If $A$ and $AB-BA$ commute, show that $AB-BA$ is nilpotent
negative_passages.docid: 57543
negative_passages.text: Show that a matrix is nilpotent.
negative_passages.docid: 2798
negative_passages.text: 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.
negative_passages.docid: 50266
negative_passages.text: In the following figure prove that: $AK||BC$
negative_passages.docid: 141173
negative_passages.text: Are there matrices such that $AB=I$ and $BA \neq I$
negative_passages.docid: 41514
negative_passages.text: Let $trcA=0$.why $A=M+N$ where $M$ and $N$ are nilpotent matrices?
negative_passages.docid: 106322
negative_passages.text: Let $A$ and $B$ be square matrices of the same size such that $AB = BA$ and $A$ is nilpotent. Show that $AB$ is nilpotent.
negative_passages.docid: 72092
negative_passages.text: If $A^2 = I$, $B^2=I$ and $(AB)^2=I$, then $AB = BA$
negative_passages.docid: 17430
negative_passages.text: If $\operatorname{rank}\left( \begin{bmatrix} A &B \\ C &D \end{bmatrix}\right)=n$ prove that $\det(AD)=\det(BC)$
negative_passages.docid: 22215
negative_passages.text: How to prove Binet theorem, namely $\det ( AB ) = \det (A ) \det ( B )$?
negative_passages.docid: 66968
negative_passages.text: if $a,b$ are nilpotent elements of a commutative ring $R$, show that $a+b$ is also nilpotent