## In this paper, we consider the problem of computing the rank of a block-structured symbolic matrix (a generic partitioned matrix) A = (Aαβxαβ), where Aαβ is a 2× 2 matrix over a field F and xαβ is an indeterminate for α = 1, 2,. 在本文中，我们考虑计算块结构符号矩阵（通用分区矩阵）A = (Aαβxαβ) 的秩的问题，其中 Aαβ 是域 F 上的 2×2 矩阵，xαβ 是 α 的不定矩阵= 1, 2,。

A combinatorial algorithm for computing the rank of a generic partitioned matrix with $$2 \times 2$$ submatrices

## In this paper, we consider the problem of computing the degree of the determinant of a block-structured symbolic matrix (a generic partitioned polynomial matrix) $$A = (A_{\alpha \beta } x_{\alpha \beta } t^{d_{\alpha \beta }})$$, where $$A_{\alpha \beta }$$ is a $$2 \times 2$$ matrix over a field $$\mathbf {F}$$, $$x_{\alpha \beta }$$ is an indeterminate, and $$d_{\alpha \beta }$$ is an integer for $$\alpha , \beta = 1,2,\dots , n$$, and t is an additional indeterminate. 在本文中，我们考虑计算块结构符号矩阵（通用分区多项式矩阵）$$A = (A_{\alpha \beta } x_{\alpha \beta } t^ {d_{\alpha \beta }})$$，其中 $$A_{\alpha \beta }$$ 是域 $$\mathbf {F}$$ 上的 $$2 \times 2$$ 矩阵，\ (x_{\alpha \beta }\) 是不确定的，而 $$d_{\alpha \beta }$$ 是对于 $$\alpha , \beta = 1,2,\dots , n$$ 的整数，并且t 是一个附加的不定数。

A combinatorial algorithm for computing the degree of the determinant of a generic partitioned polynomial matrix with 2×2 submatrices