query: Prove $\det(I_m + AB) = \det(I_n + BA)$.
positive_passages.docid: 88349
positive_passages.text: Are the eigenvalues of $AB$ equal to the eigenvalues of $BA$? (Citation needed!)
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: 115582
negative_passages.text: Prove that if a product$ AB$ of $n\times n$ matrices is invertible, so are the factors $A$ and $B$.
negative_passages.docid: 75855
negative_passages.text: Show that $\det{(A + sxy^*)}$ is linear in $s$
negative_passages.docid: 106511
negative_passages.text: Determinant of the sum of matrices: $\det (A + B^T) = \det(A^T + B)$
negative_passages.docid: 104258
negative_passages.text: Proof Verification: $\dim\mathrm{Im} T+\dim\ker T=\dim V$
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: 137422
negative_passages.text: $\mathrm{card}(\mathbb{Z}^n/M\mathbb{Z}^n) = |\det(M)|$?
negative_passages.docid: 64742
negative_passages.text: Prove a residue matrix $A$ (with coefficients in $\mathbb Z_n)$ has an inverse if and only if $\gcd(\det A,n) = 1$
negative_passages.docid: 82925
negative_passages.text: Let the matrix $A=[a_{ij}]_{n×n}$ be defined by $a_{ij}=\gcd(i,j )$. How prove that $A$ is invertible, and compute $\det(A)$?
negative_passages.docid: 107494
negative_passages.text: prove $\frac {ab}{a+b} \geq \sum ^n_{i=1} \frac{a_ib_i}{a_i+b_i}$
negative_passages.docid: 75002
negative_passages.text: show that $\det(A)=0$ in this case
negative_passages.docid: 103061
negative_passages.text: Why is it true that $\mathrm{adj}(A)A = \det(A) \cdot I$?
negative_passages.docid: 13026
negative_passages.text: Prove that if AB=I_m, then rk(A)=rk(B).
negative_passages.docid: 10083
negative_passages.text: Determinant of matrix times a constant.
negative_passages.docid: 71442
negative_passages.text: How to prove $\det(I+uv^\intercal)=1+v^\intercal u$
negative_passages.docid: 81181
negative_passages.text: Prove that $\det(M-I)=0$ if $\det(M)=1$ and $MM^T=I$