# Understanding $\,\det(A+B)$

From this paper on Determinant of sums, where $$$$\det(A+B) = \sum_{r} \sum_{\alpha,\beta} (-1)^{s(\alpha) + s(\beta)} \det(A[\alpha|\beta])\det(B[\alpha|\beta]),$$$$ the meaning of $$A[\alpha|\beta]$$, and how the sum runs over $$\alpha, \beta$$ is not clear to me. Would appreciate an explanation of this this result.

• Page $3$ (or page $131$) of your paper already explained what these symbols mean after display your posted equality (equality $(1)$ in the paper). Jan 10, 2023 at 14:08
• It talks of $\alpha, \beta$ being sequences! Could you illustrate it with an example, when say $n=4$? Jan 10, 2023 at 14:11
• $\alpha$ and $\beta$ are increasing sequences of length $r$: $\alpha=(\alpha_1,\alpha_2,...,\alpha_r)$ and $\beta=(\beta_1,...,\beta_r)$, with $a_i$ and $\beta_i$ strictly increasing and drawn from the set $\{1,2,...,n\}$. Mark the rows $\alpha_1,...,\alpha_r$ and the columns $\beta_1,...,\beta_r$ of $A$. Those rows and columns intersect at $r^2$ elements. The matrix formed by those elements is $A[\alpha|\beta]$. The matrix $B[\alpha|\beta]$ is similar, but using the rows that are not in $\alpha$ and the columns that are not in $\beta$.
– plop
Jan 10, 2023 at 14:13
• For $n=4$, $r=2$, the sum over $\alpha, \beta$ is the sum over ordered pairs of $(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)$, $30$ terms in total. Jan 10, 2023 at 14:14
• You mean, for $n=4$, we have $r=0,1,2,3,4$, and it is for $r=2$ we sum over the ordered pairs you mentioned? What about $r=0$ and $r=1$? Jan 10, 2023 at 14:21
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