query: Prove $\det(I_m + AB) = \det(I_n + BA)$.
positive_passages.docid: 12355
positive_passages.text: Sylvester's determinant identity
negative_passages.docid: 65783
negative_passages.text: Proving: "The trace of an idempotent matrix equals the rank of the matrix"
negative_passages.docid: 13026
negative_passages.text: Prove that if AB=I_m, then rk(A)=rk(B).
negative_passages.docid: 90486
negative_passages.text: If $(A-B)^2=O_2$ then $\det(A^2 - B^2)=(\det(A) - \det(B))^2$
negative_passages.docid: 21788
negative_passages.text: Proving that if $ad-bc = \pm 1$, then $\gcd(x,y) = \gcd(ax +by, cx + dy)$
negative_passages.docid: 81181
negative_passages.text: Prove that $\det(M-I)=0$ if $\det(M)=1$ and $MM^T=I$
negative_passages.docid: 5183
negative_passages.text: Divisibility of determinant.
negative_passages.docid: 101040
negative_passages.text: Prove the equality: $\det\left[\begin{smallmatrix} -2a &a+b &a+c \\ b+a&-2b &b+c \\ c+a&c+b &-2c \end{smallmatrix}\right] = 4(a+b)(b+c)(c+a)$
negative_passages.docid: 109745
negative_passages.text: If $(a,k)=(b,m)=1$, prove that $(ab,km)=(a,m)(b,k)$.
negative_passages.docid: 11557
negative_passages.text: How to prove $\det(e^A) = e^{\operatorname{tr}(A)}$?
negative_passages.docid: 23847
negative_passages.text: Determinant of a matrix whose $(i,j)$th element is $|i-j|$?
negative_passages.docid: 124868
negative_passages.text: Prove that $\frac{bc}{b+c}+\frac{ca}{c+a}+\frac{ab}{a+b}<\frac{a+b+c}{2}$
negative_passages.docid: 63119
negative_passages.text: Let $M=\big( \begin{smallmatrix} A & B \\ C & D \end{smallmatrix} \big)$. Prove $\det(M)=\det(A)\cdot \det(D-C·A^{-1}·B)$.
negative_passages.docid: 127085
negative_passages.text: Bound on the difference of two determinants
negative_passages.docid: 110703
negative_passages.text: How Can I prove (a+b) mod m = (a mod m) + (b mod m)) mod m?
negative_passages.docid: 11895
negative_passages.text: If $AB=0$ prove that $\mathrm{rank}(A)+\mathrm{rank}(B)\leq n$
negative_passages.docid: 6410
negative_passages.text: Prove that $(ma, mb) = |m|(a, b)\ $ [GCD & LCM Distributive Law]