Consider a block matrix of the form $$ A = \left[ \begin{array}{cc}A_1 & tB_1 \\ tB_2 & A_2\end{array}\right],$$ where $t>0$, $A_i$ are square matrices and $B_i$ are matrices of the transpose size. If I'm not mistaken, there is a result that states that for sufficiently small $t$ then the matrix $A$ is invertible under some conditions on the data, so I want to find a reference for such result.
I want to know how bounded in terms of the data of $A_i$ and $B_i$ the value of $t$ must be and also what are the minimal hypotheses we need to have on $B_i$ and $A_i$ for such result to work. Where can I find more about this result?
This claim holds iff. $A_1,A_2$ are invertible. Why? This follows easily from the continuity of the determinant. As $t\to0$, $A$ tends to the diagonal block matrix with entries $A_{1,2}$ and that has determinant $\det(A_1)\det(A_2)\neq0$ iff. $A_1,A_2$ are both invertible. And if $\det(A_1)\det(A_2)\neq0$ then by continuity $\det A\neq0$ for small $t$...
Given $A_1,A_2$ are invertible we may compute: $$\det A=\det(A_1)\cdot\det(A_2-t^2B_2(A_1)^{-1}B_1)=\det(A_1A_2)\cdot\det(I-t^2(A_2)^{-1}B_2(A_1)^{-1}B_1)$$Using $\det(I+X)=1+\operatorname{tr}(X)+o(\|X\|)$ as $X\to0$ we can infer that this quantity is $\det(A_1A_2)+\mathcal{O}(t^2)$ as $t\to0$.
I wouldn't know how to give more precise bounds.