query: If $B$ is nilpotent and $AB=BA$ then $\det(A+B) = \det(A)$
positive_passages.docid: 142980
positive_passages.text: If $\,A^k=0$ and $AB=BA$, then $\,\det(A+B)=\det B$
negative_passages.docid: 103061
negative_passages.text: Why is it true that $\mathrm{adj}(A)A = \det(A) \cdot I$?
negative_passages.docid: 62474
negative_passages.text: If $AB=I_n$ and $BA=I_m$ then $n=m$.
negative_passages.docid: 54479
negative_passages.text: If $x,y$ are nilpotent and commute, $x+y $ 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: 13026
negative_passages.text: Prove that if AB=I_m, then rk(A)=rk(B).
negative_passages.docid: 128820
negative_passages.text: Prove that if $\gcd(a,b)=1$ then $\gcd(ab,c) = \gcd(a,c) \gcd(b,c)$
negative_passages.docid: 7034
negative_passages.text: Prove that $A+I$ is invertible if $A$ is nilpotent
negative_passages.docid: 128760
negative_passages.text: If $A$ and $B$ two $n\times n$ matrix such that $AB=BA$, then $\dim(A\cap B)$?
negative_passages.docid: 141173
negative_passages.text: Are there matrices such that $AB=I$ and $BA \neq I$
negative_passages.docid: 50266
negative_passages.text: In the following figure prove that: $AK||BC$
negative_passages.docid: 78224
negative_passages.text: Suppose $A$ is a general $n \times n$ matrix and $B$ is obtained by interchanging two rows of $A$. Prove that $\det(B) = -\det(A)$
negative_passages.docid: 57543
negative_passages.text: Show that a matrix is nilpotent.
negative_passages.docid: 111824
negative_passages.text: Sum of nilpotent and element is a unit in ring?
negative_passages.docid: 83475
negative_passages.text: Prove Or Disprove: tr(AB)=tr(A)*tr(B)
negative_passages.docid: 55056
negative_passages.text: To show: $\det\left[\begin{smallmatrix} -bc & b^2+bc & c^2+bc\\ a^2+ac & -ac & c^2+ac \\ a^2+ab & b^2+ab & -ab \end{smallmatrix}\right]=(ab+bc+ca)^3$
negative_passages.docid: 55062
negative_passages.text: $2\times 2$ matrix $C=AB-BA$