# Questions tagged [laplace-expansion]

Laplace expansion is a method for expanding determinants in terms of minors, determinants of some related smaller matrices.

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### Intuition for Laplace expansion

I've been trying to look for an intuitive understanding for the Laplace expansion of the determinant. I first tried looking for the proof but let's just say it was way to complicated for my ...
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### Spherical multipole moments after flipping an axis

I have an interior spherical multipole expansion (as in Modern Electrodynamics by Andrew Zangwill): $$f(\textbf{r}):=\sum_{l=0}^{\infty} \sum_{m=-l}^{l} B_{lm} r^{l} Y_{lm}^*$$ with spherical ...
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### Matrix determinant where $a_{ij} = i + j$ [duplicate]

So I'm studying for my course of linear algebra and the following problem was inside the book with exercises. "Given a matrix $A \in \mathbb{R}^{n \times n}$, where $a_{ij} = i + j$, calculate it'...
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### Proving:$\operatorname{Proj}_{U^\perp}(x)=-\frac1{\det(A^TA)} X(u_1,\ldots, u_{n-2}, X(u_1,\ldots, u_{n-2}, x))$

The problem I'm trying to solve is as follows, which was posed to me by my professor as an exercise: Let $x, u_i \in \Bbb R^n$, $A = (u_1, u_2, \ldots, u_{n-2})$ and $\{u_1, u_2, \ldots, u_{n-2}\}$ ...
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### How can I prove that the determinant of the matrix is $0$ only when $a=b=c=d=0$? [duplicate]

How can I prove that the determinant of the matrix is $0$ only when $a=b=c=d=0$? I tried with the Laplace Method of Expansion but I cannot solve the final equation. \begin{bmatrix} a & b ...
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### Prove that the Laplace expansion for the determinant is the same for any choice of row or column

So I understand from the definition of the determinant that: $$\det(A)=\sum_{i}^{} (-1)^{i+k}a_{ki}M_{ki}$$where we define $M_{ki}$ to be the determinant of an $(n-1) \times (n-1$) matrix formed by ...
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### Characteristic polynomial problem

$$\begin{pmatrix} 1 & \cdots & n \\ n+1 & \cdots & 2n \\ \vdots & \ddots & \vdots \\ n^2-n+1 & \cdots & n^2 \end{pmatrix} .$$ I am trying to find ...
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