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I have trouble on finding the inverse matrix of $V$, given below. I tried finding first the inverse matrix to the case where $n=2,3$. But, I cant find a pattern that will lead me to the general one. Now, I am stuck at the case where $n=4$.

$$V= \displaystyle{\begin{bmatrix} \sum_{1 \neq k} m_{1k} +(\gamma_1+\rho_1\tau_1) & -m_{21} & \cdots & -m_{n1} & 0 & 0 & \cdots & 0 \\ -m_{12}& \sum_{2 \neq k} m_{2k} +(\gamma_2+\rho_2\tau_2) & \cdots & -m_{n2} & 0 & 0 &\cdots & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ -m_{1n}&-m_{2n}& \cdots & \sum_{n \neq k} m_{nk} +(\gamma_n+\rho_n\tau_n)& 0 &0&\cdots &0 \\ -\rho_1\tau_1 & 0 & \cdots & 0 & \delta_1 &0 &\cdots& 0\\ 0 &-\rho_2\tau_2& \cdots & 0 & 0 &\delta_2 &\cdots& 0 \\ \vdots & \vdots & \ddots&\vdots&\vdots&\vdots& \ddots&\vdots \\ 0 & 0 & \cdots & -\rho_n\tau_n & 0 &0 &\cdots& \delta_n \end{bmatrix}}$$

Another question: Is there any result that will help me, if I get the determinant of matrix $V$, can I get the inverse matrix of $V$? Because there is a result here Determinant of a block lower triangular matrix that gives the determinant of an $n$x$n$ matrix. So maybe this will help me. Thank you.

jiid
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    https://math.stackexchange.com/questions/2316555/the-inverse-of-a-block-upper-triangular-matrix – Daniel Apr 30 '21 at 05:18
  • thanks sir. does this apply to block lower triangular matrix right? – jiid Apr 30 '21 at 05:22
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    The idea yes. Of course, the formula will be slightly different (concretely, the order of the multiplication is going to be the opposite one). Just figure it out the same way as they did in the answer. – Daniel Apr 30 '21 at 05:27

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