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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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Laplace’s method for asymptotic series expansion [closed]

I have this integral for which I need to calculate the leading asymptotic behavior: $$ I\left(x\right)=\int_0^{\frac{\pi}{2}}e^{-x\tan\left(t\right)}dt,\quad x\rightarrow \infty$$ $\phi\left(t\right)=\...
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Stop criterion for an iterative solver for laplace equation

Suppose I am iterating with Euler's forward method on finite differences like so $$ z_{n+1}(k, l) = \begin{cases} z_n(k, l) + \frac{1}{4}\left(z(k - 1, l) + z(k, l -1) -4z(k, l) + z(k, l+1) + z(k + 1, ...
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Determinant of a matrix is equal to the determinant of its transpose

For a square matrix A, I want to show that $$ | \mathbf{A}^T| = |\mathbf{A}| $$ Proof: Let the same permutation that changes $ \varphi(j_1,...,j_n) $ into $ \varphi(1,...,n) $, change $ \varphi(1,...,...
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Laplace expansion vs Cofactor expansion? [closed]

I would like to know the difference between cofactor expansion and Laplace expansion. It looks like they are the same thing under different names. However, I was told that Laplace expansion is more ...
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Classical reference for “vector-valued” Laplace expansion?

In relation to the $2\times 2$ (say) determinant $$\begin{vmatrix}a&b\\c&d\end{vmatrix},$$ consider the “bordered” determinant $$\begin{vmatrix} x&a&b\\ x&a&b\\ y&c&d \...
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How can I solve Laplace Tranformation of $1/s^{5/2}$?

I have just started Laplace Transformation And I came across a problem which contains $1/s^{5/2}$ How to solve it? I know $\mathcal L\{t^n\}= n!/s^{n+1}$ Please say how to solve it.
mainak mukherjee's user avatar
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Is there an easy way for a person to compute the determinant of an arbitrary matrix?

I'm having a tough time proving statements that involve the determinant of an arbitrary matrix and was wondering if there is just an easier way to compute it or some simpler equivalent definition for ...
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prove the determinant identities

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\...
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how to solve this linear system of three equations using Cramer's rule?

I have a 3-by-3 matrix, A=$\left [ \begin{matrix} 1 & 2 & 3 \\ 1 & 0 & 1 \\ 1 & 1 & -1\\ \end{matrix} \right]$ the known terms are (-6, 2, -5), at the right ...
Gabriel Burzacchini's user avatar
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what's wrong with this matrix (to find the determinant using Laplace expansion)?

I have to compute the determinant of this 4x4 matrix: \begin{bmatrix}2&1&3&0\\-1&0&1&2\\2&0&-1&-1\\-3&1&0&1\end{bmatrix} this is what I did: I swapped ...
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Proof of the Laplace Expansion? [closed]

I just learned about Laplace Expansion for determinant calculation in high-school, they taught me how to calculate minors cofactors and everything but they did not include the proof in the book. I ...
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Determinant of upper triangular matrix with first column not $0$

I have a Matrix $A \in \mathbb{R}^{n \times n}$ which is of the form $A$ = \begin{pmatrix} a_{11} & a_{12} & a_{13} & ... & a_{1n}\\ a_{21} & a_{22} & a_{23} & ... & a_{...
Maximilian's user avatar
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Asymptotic expansion for the following integral

I was trying to find the asymptotic expansion for $$\int_0^1 \sqrt{t(1-t)}(t+a)^{-x} \; \mathrm{d}t,$$ for $a>0$ as $x \rightarrow \infty$. I have already tried re-writing $$(t+a)^{-x}=\exp(- x\log(...
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Matrix, cofactor, minor, Determinant proof

I'm looking at Apostol's Calculus and I found something that might be trivial but I can't see it directly. $det A_{12}'=\begin{vmatrix} 0 & 1 & 0 & ...& 0\\ a_{21} & 0 & a_{23} ...
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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'...
Matthias K.'s user avatar
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two simple laplace transformation with simple functions

this is a simple laplace transformation can someone please help me solve this I am stuck
bendover's user avatar
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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 $\{u_1, u_2, \ldots, u_{n-2}\}$ ...
learning_linalg's user avatar
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Determinant of matrix transpose and Laplace expansion, application

I recently read this paper in which the authors construct a matrix related to the Collatz conjecture such that $$ m_{ij} = \begin{cases} 1 \text{ if } i = j\\ x \text{ if } c(i) = j\\ 0 \...
humblebee's user avatar
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Determine the determinant of a companion matrix

Calculate for $ n \geq 2 $ and $ x, a_{0}, a_{1}, \ldots, a_{n-1} \in \mathbb{R} $ the determinant of the following matrix: $$\begin{bmatrix} {x} & {0} & {\cdots} & {\cdots} & {\cdots}...
Ludwig von Drake's user avatar
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Monic polynomial and companion matrix

Problem Let $p(T) := T^n-\alpha_{n-1}T^{n-1}-\alpha_{n-2}T^{n-2}-\cdots-\alpha_0 \in K[T]$. Additionally we have the companion matrix of $p$ $$A:= \begin{bmatrix} 0 & 1 & 0 & 0 &...
navix98's user avatar
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Determinant of 4x4 w/ all entries unknowns

Any help with this problem would be greatly appreciated. If $A$ is the matrix $$ A = \begin{bmatrix} a & b & c & d+1\\ a & b & c+1 & d \\ a & b+1 & c ...
user734624's user avatar
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Why is the determinant $1$ not $-2$

I am trying to find the determinant of this matrix. Let $A$ be the matrix : \begin{bmatrix}2&2&1\\1&0&5\\1&1&0\end{bmatrix} Using row operations, I can change $R_3$ to ( $-...
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Intuition of finding the formula of Laplace's expansions of the Determinant?

How did Laplace find the formula $\left |A \right |=\sum_{i=1}^{n}(-1)^{i+j}(A)_{ij}M_{ij}$? What is the intuition of the evalution of this formula? Note: I'm not asking for proof that the formula ...
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How to find a multiple eigenvalue of a square matrix?

$$ \ A= \begin{bmatrix} 2 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 3 & 5 & -2 & -2 \\ -2 & 3 & 3 & 5 \\ \end{bmatrix} $$ For this matrix, I am supposed to ...
camtexem's user avatar
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$AC^T = \det(A)I$

Let A = $\begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} &...
Kid Cudi's user avatar
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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 ...
Oliver Smith's user avatar
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2 answers
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Inverse Laplace transformation that is slightly different from known transformation.

I have the following inverse laplace transformation: $L^{-1} =\frac{s}{(s-3)(s-4)(s-12)}$ After looking at the laplace transformations the closest I've found is: $\frac{ae^{at}-be^{bt}}{a-b} = \...
MKUltra's user avatar
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Laplace Transform for an Initial Value Problem

I think I have this one all figured out, I just need someone to tell me if I got it or not... so I have: $$ \begin{cases} 9y''+12y'+4y = e^{-2t}\\ \\ y(0)=y'(0)=1 \end{cases} $$ so using the ...
George Dimitriou's user avatar
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5 answers
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How to compute the determinant of this Toeplitz matrix?

Given a positive integer $n$, express$$ f_n(x) = \left|\begin{array}{c c c c c} 1 & x & \cdots & x^{n - 1} & x^n\\ x & 1 & x & \cdots & x^{n - 1} \\ \vdots & x &...
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How to calculate the determinant?

I need help with this determinant ($n \times n)$: $D_n = \begin{vmatrix} a & x & x & \dots & x & x \\ y & a & x & \dots & x & x \\ y & y & a & \...
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Assume that $f(t)$ and $tf(t)$ are bounded. $F(s)=\mathcal{L}(f(t))(s)$, show that $\mathcal{L}(tf(t))(s)=-F'(s)$.

Question : Assume that $f(t)$ and $tf(t)$ are bounded. Denoting $$F(s)=\mathcal{L}(f(t))(s)$$ show that $$\mathcal{L}(tf(t))(s)=-F'(s)$$ My Try : I know that $$\mathcal{L}(f(t))(s)=\int_0^{\infty}e^{-...
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Proof regarding how the determinant changes in a special matrix after replacing a row with an 'all--one' row

The actual problem: In each row of an $n \times n$ matrix, the sum of elements equals $2008$. Prove that if we replace all values of one row by $1$-s, the determinant becomes $\frac{1}{2008}$ times ...
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How would I find the determinant of the following $4 \times 4$ matrix?

\begin{pmatrix}-5&0&0&0\\ \:5&4&0&0\\ \:-1&8&3&0\\ \:-6&-3&3&1\end{pmatrix} I know that if it was a $3 \times 3$ matrix, I could simply do the Laplace ...
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Is there an official reference for block-by-block Laplacian expansion?

Consider a $tn\times tn$ matrix $M$. We can see this matrix as a n $n\times n$ matrix of $t\times t$ blocks. It is known that if all blocks commute, than you can compute the determinant $D$ of $M$ in ...
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A question on the derivation of Jacobi's formula

In the standard derivation of Jacobi's formula One uses the Laplace expansion of the determinant of some matrix $A$ : $$\text{det}\,A=\sum_{j}A_{ij}C_{ij}$$ where $i$ is a fixed, but arbitrary row, ...
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Why are $\cosh$ and $\sinh$ used in solving this case of Laplace's equation?

I'm currently trying to find the separable solution to the PDE $$ u_{xx} + u_{yy} = 0 $$ Subject to the boundary conditions $$ u(0,y) = u(a,y) = 0 $$ To attempt this, I have expressed $u(x,y) = X(x)Y(...
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4 votes
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Calculating general $n\times n$ determinant

I'm given determinant $\begin{vmatrix} 1 & 2 &3 & \cdots & n -1 & n \\ 2 & 3 &4 & \cdots & n & 1 \\ 3 & 4 &5 & \cdots & 1 & 2 \\ \vdots &...
Accelerate to the Infinity's user avatar
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1 answer
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Multipole expansion of Coulomb function in 2D (Laplace expansion)

In 3D, Coulomb function in Cartesian coordinates $(x, y, z)$ reads $$\frac{1}{|\boldsymbol{x}-\boldsymbol{x'}|} = \frac{1}{\sqrt{(x-x')^2 + (y-y')^2 + (z-z')^2}}$$ which may be expanded in spherical ...
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Given a tridiagonal Matrix $K_n \in \mathbb{R}^{n\times n}$ with specific elements, show that det$(K_n) = n+1$

Let $K_n \in \mathbb{R}^{n \times n}$, specifically for general $n \in \mathbb{N}$ $$ (K_n)_{ij} = \begin{cases} 2&, i=j\\ -1&, \|i-j\| = 1\\ 0 &, \text{otherwise} \end{cases} $$ where $...
oscurecer's user avatar
1 vote
1 answer
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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 ...
Elk's user avatar
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2 votes
1 answer
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Is there a simple way of calculating determinant of "reverse arrowhead" matrix?

I have some problems with finding the determinant of the following matrix. I have tried a simple Laplace expansion on the first row, but I have a feeling I am missing some simple trick here. $$ \begin{...
zhero's user avatar
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2 answers
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Finding eigenvalues of a $3\times3$ matrix with Laplace expansion

Currently working on problem for a linear algebra class, but having a difficult time grasping eigenvalues. Here are the steps I'm doing: $$A=\begin{bmatrix}-5 & 1 & 0 \\ 0 & -4 & 3 \\ ...
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3 answers
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Laplace Question $f(t) = e^{-t} \sin(t)$

I need help with this Laplace question. $$f(t) = e^{-t} \sin(t) $$ Answer should be $\dfrac{1}{s^2 + 2s + 2}$ What I'm currently doing is as follows: $u = \sin(t)\qquad$ $dv = e^{-(s+1)t}dt$ $...
Ricardo Castillo's user avatar
6 votes
2 answers
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Help me with the result of this determinant..

$$ D = \begin{vmatrix} 1 & 1 & 1 & \dots & 1 & 1 \\ 2 & 1 & 1 & \dots & 1 & 0 \\ 3 & 1 & 1 & \dots & 0 & 0 \\ \vdots & \vdots & \...
A6EE's user avatar
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11 votes
2 answers
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Proof of Leibniz formula from Laplace expansion

I'm trying to prove Leibniz formula for the determinant using Laplace expansion. Here's my attempt: For a $1 \times 1$ matrix $A = \begin{pmatrix}a_{11}\end{pmatrix}$, define $\det A = a_{11}$. For ...
David's user avatar
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1 answer
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Super Simple Proof of Cofactor Expansion

Is it possible to provide a super simple proof that cofactor expansion gives a determinant value no matter which row or column of the matrix you expand upon? E.g., super simply prove that $$\det(A) =...
John Doe's user avatar
2 votes
4 answers
2k views

How to compute determinant of $n$ dimensional matrix?

I have this example: $$\left|\begin{matrix} -1 & 2 & 2 & \cdots & 2\\ 2 & -1 & 2 & \cdots & 2\\ \vdots & \vdots & \ddots & \ddots & \vdots\\ 2 & 2 &...
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