3 problems

Let $A_i\in\mathcal F^{m_{i+1}\times m_i}$ for $i=1,2,\ldots,k$, and let $A_i^-$ denote an inner inverse satisfying $A_iA_i^-A_i=A_i$. Conjecture 30.14. The minimum rank of the pro…

Let $A+iX+jY+kZ$ be a quaternion matrix, where $i^2=j^2=k^2=-1$ and $ijk=-1$, with real matrices $A,X,Y,Z$. Conjecture 29.7. The supplied statement asserts a formula for … but the…

Let $A\in\mathcal F^{m\times m}$ be a square matrix, and let $A^-$ be a generalized inverse of $A$. Conjecture 21.20. For every choice of $A^-$, the rank of the commutator … is eve…