prove the determinant identities

Asked Dec 28 '21 at 13:48

Active Dec 28 '21 at 14:57

Viewed 190 times

Let $M$ be an $n \times n$ matrix, $N = \{1, \dots , n \}$ and $V \subseteq N$. Let $M^{(V)}$ denote the submatrix $(m_{i,j})$ of $M$ with $i,j \in V$. Prove that

$$\det(Y-\iota I_n)^{(U)}=\sum_{W\subseteq U} (-\iota)^{card(W)} \det(Y^{(U-W)})$$ where $U \subseteq \{1, \dots , n\}$ https://en.wikipedia.org/wiki/Laplace_expansion
If $Y$ is a regular(invertible) ($n \times n$)-matrix over some arbitrary field $K$ and $U \subseteq \{1, \dots , n\}$ and $\bar U=\{1, \dots , n\}-U.$ then

$$\det(Y^{(U)})=\det(Y^{-1})^{(\bar U)} \det(Y)$$ I don't know how to prove these statements, please help!

edited Dec 28 '21 at 14:57

Rodrigo de Azevedo

23,223

asked Dec 28 '21 at 13:48

Manish Saini

1

Try to visualize the problem in $\mathbb R^3$ in order to understand the sens of it. The determinant is an alternating multilinear map so it will be useful for you to know some key definitions on this topic. Your problem is one of multilinear algebra. – Ataulfo Dec 28 '21 at 14:04
1

@Piquito Thanks for the hint, can you suggest some references on multilinear algebra? – Manish Saini Dec 28 '21 at 14:07
only in French, I am sorry. In Google you can find a lot I guess. – Ataulfo Dec 28 '21 at 14:18
@Piquito Okay Thanks! – Manish Saini Dec 28 '21 at 14:27
2

For part 2: https://mathoverflow.net/questions/87877/jacobis-equality-between-complementary-minors-of-inverse-matrices – darij grinberg Dec 28 '21 at 15:00
1

For part 1: e.g., Proposition 6.4.29 in my An Introduction to Algebraic Combinatorics as of October 14, 2021. (Set $x = -\iota$.) – darij grinberg Dec 28 '21 at 15:02
@darijgrinberg, Thanks for these useful references. – Manish Saini Dec 29 '21 at 08:44
@darijgrinberg I have one doubt related to part 1; in proposition 6.4.29 if set $x=-\iota$ then it will equals to $det(A-\iota I_n)$ not $det (A-\iota I_n)^{(U)}$ ?? – Manish Saini Dec 29 '21 at 10:34
Oh, right -- you also need to apply the proposition to $A^{(U)}$ instead of $A$. – darij grinberg Dec 29 '21 at 11:28

prove the determinant identities

0 Answers0