For questions on or related to the technique of Gaussian elimination, used in solving systems of linear equations.

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Count of solutions to matrix equations

Given these modular equations: $$a_{1,1} x_1 + a_{1,2} x_2 + \cdots + a_{1,n} x_n = b_1 \bmod p $$ $$a_{2,1} x_1 + a_{2,2} x_2 + \cdots + a_{2,n} x_n = b_2 \bmod p $$ $$\vdots$$ $$a_{m,1} x_1 + ...
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1answer
29 views

Show that if the leading principal minors of a nonsingular $n\times n$ matrix $A$ are all nonzero then the matrix $A$ has $LU$ factorization

I am stucked at this problem: Prove by induction that if the leading principal minors of an $n\times n$ nonsingular matrix $A$ are all nonzero then the matrix $A$ has $LU$ factorization. (The ...
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74 views

Number of Arithmetic Operations in Gaussian-elimination/Gauss-Jordan Hybrid Method for Solving Linear Systems

I am stucked at this problem from the book Numerical Analysis 8-th Edition (Burden) (Exercise 6.1.16) : Consider the following Gaussian-elimination/Gauss-Jordan hybrid method for solving linear ...
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1answer
22 views

How many solutions depending on the parameter (augmented matrix?)

I have to find how many solutions have got the following equations, depending on p parameter? $ \begin{bmatrix} 5 & p & 5 \\ 1 & 1 & 1 \\ p & p & 2 \end{bmatrix} $ $ ...
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1answer
13 views

Gaussian elimination and pivots

I'm having trouble answering the question posed below ("Question: Which x on the left..."). It seems like there's only 2 options as the first two along the diagonal are already boldfaced. I'm not ...
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2answers
19 views

matrix elementary column operations

Till now i was using the elementry row operations to do the gaussian elemination or to calculate the inverse of a matrix. As i started learning the Laplace's transformation to calculate the ...
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2answers
42 views

Solution of System of linear equations

I have three equations: $$ \begin{cases} 4y + z = 2\\ 2x + 6y - 2z = 3\\ 4x + 8y - 5z = 4 \end{cases} $$ Applying Gauss elimination I get: $$ \left[ \begin{array}{ccc|c} ...
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Can I go from the LU factorization of a symmetric matrix to its Cholesky factorization, without starting over?

I mistakenly computed the LU factorization and then realized that the question is asking for a Cholesky factorization, i.e., finding a lower triangular matrix L such that the symmetric matrix A has ...
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3answers
54 views

Why doesn't Gaussian elimination change the solution set?

Of course, Gaussian elimination is safe to use as proven by the countless systems I've solved with it while practising my linear algebra (which is I must add very basic and low-level), but when Jim ...
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1answer
22 views

Gaussian Elimination General Solution

Find the general solution of the following system of equations: Using Gaussian Elimination I was able to get the following solutions for these equations: x = 2 y = 1 z = 0 However, this is not ...
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1answer
38 views

gauss jordan matrix involving parameter $k$

Could anyone help me in solving this matrix? $$\left[\begin{array}{ccc|c} k+2& k-1& k& 2\\ 0& k+2& 2& 0\\ 0& 0& k^2+k-2& k+2 ...
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0answers
21 views

Incomplete Gauss Jordan elimination: what have I left out

I wrote some C++ to implement a Gauss-Jordan elimination: it's all a bit rough and ready (I wanted to get a fix on the algorithm rather than write a piece of "production" code) but you can see it ...
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1answer
34 views

How to solve system of equations with fractions?

I have this equation $U=\frac{a_0+a_1X+a_2Y}{1+c_1X+c_2Y}$ $V=\frac{b_0+b_1X+b_2Y}{1+c_1X+c_2Y}$ Where (X,Y) is a point on a quadrilateral and (U,V) is a point on a rectangle, and (X,Y) actually ...
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1answer
28 views

Reducing a Matrix with Guass-Jordan elimination

I am doing some homework, but I am stuck on this problem. $-x+y=-2$ $-3x+2z=13$ $2x-2z=-6$ Here's my work: That wasn't accepted. I then threw it in matlab, to try to work my way ...
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2answers
59 views

For what values of $a$, $b$, and $c$ the above system has: One solution. Infinitely many solutions. No solutions.

I am stuck with this now, I tried reducing the matrix to row echelon form, but it gets a bit hard. Is there not a simpler way? The system is: \begin{align*} a x + b y − 3 z &= −3\\ −2 x − b y + ...
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0answers
24 views

How to properly detect rows to be swapped in a Gaussian elimination?

I'm trying to describe an algorithm for solving solvable linear systems. The Gaussian elimination is pretty straightforward in terms of adding multiples of rows. However, consider the following ...
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2answers
18 views

System of non-homegeneous linear equations

I need to find a relation between $a$, $b$, $c$, $d$ in order the system with the following augmented matrix has at least one non-trivial solution. I have tried both the Gaussian and Gauss-Jordan ...
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14 views

Online Gaussian-elimination solver that produces Latex code of the detailed solution.

I wonder if there exists an online (free) Gaussian-elimination solver that generates a $\LaTeX$ code describing each steps of the detailed solution, i.e. telling which elementary operation is ...
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67 views

Gauss elimination: Difference between partial and complete pivoting

I have some trouble with understanding the difference between partial and complete pivoting in Gauss elimination. I've found a few sources which are saying different things about what is allowed in ...
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2answers
238 views

How exactly do we do Gauss elimination?

This is a matrix: $$\begin{bmatrix} 1 & 1 & 1\\ 1 & 2 & 3\\ 1 & 3 & k \end{bmatrix}\begin{bmatrix} x\\ y\\ z \end{bmatrix}= \begin{bmatrix} 3\\ 6\\ 4+k \end{bmatrix}$$ ...
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starting with $n$ equations with $n$ unknowns, are ALL inconsistent systems of the form (e.g) 0=4?

I'm doing a program for Gaussian elimination, and I've used the fact that after elimination, if an matrix has a row of 0's with its last column not equal to 0, then the system is inconsistent. I've ...
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0answers
14 views

Generalizing the algebraic nature of linear inconsistent system

I want to know whether solving any inconsistent linear system in any dimension end up in a similar manner that is $0 = a$ where a is non zero. For 2D, the only reason for inconsistent system can be ...
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2answers
33 views

How to determine the transition matrices when doing Gaussian elimination?

Guassian elimnation can be done by swapping or adding rows, and by multiplying rows by scalars, etc. We use it to bring for example our original matrix to an upper triangular matrix, so that we can ...
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116 views

Scale vector in scaled pivoting (numerical methods)

In the scaled pivoting version of Gaussian elimination, you exchange rows/columns not only based on the largest element to be found, but rather the largest relative to the entries in its row. You ...
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2answers
38 views

Is set of three linear equations with three unknown solvable?

I have the following set of linear equations with the unknowns $h, n, i$ which I would like to express as a function of my known quantities, $e, f, g$: $$ e = h - n\\ f = h - i\\ g = i -n $$ with ...
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1answer
25 views

Gaussian elimination problem

$$x_1 + 10x_2 − 3x_3 = 8$$ $$x_1 + 10x_2 + 2x_3 = 13$$ $$x_1 + 4x_2 + 2x_3 = 7$$ when making 2nd and 3rd 1st columns 0 using Gaussian elimination, the second row second column also becomes zero, so ...
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0answers
26 views

Gauss Jordan Elimination different answers

I have a question regarding Gauss Jordan Elimination. I have this matrix: \begin{bmatrix}2&1&5&0\\1&0&-3&1\\7&2&2&1\end{bmatrix} So at the start I will switch R1 ...
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1answer
10 views

Different results while calculating eigenvectors with Gaussian elemination

Regarding this matrix $\begin{matrix} 1 & 1 \\ 1 &-1 \\ \end{matrix}$. In the end I have to solve this equation system: $(\sqrt2-1)x_1-x_2=0$ $-x_1+(\sqrt2+1)x_2=0$ While the ...
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1answer
26 views

Why do the 1's in Gauss Jordan RREF need to be along main diagonal and not other diagonal?

I've practiced G-J elimination and understand most of the algorithm insofar as it represents the different manipulations one can apply to a system of equations. However, when we're talking about ...
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1answer
49 views

Finding the y-coordinate of the peak in a gaussian distribution?

First off all, my general understanding of gaussians is not very good, and I'm having issues getting my head around this because I cannot find an explanation of them I can understand. I'm working ...
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2answers
51 views

Propositions of elementary matrix

i'm trying to solve a question about elementary matrix. When given $A_{m,n}$ and $B_{n,p}$ which differ from the Zero matrix. Also, multiplying of $A$ and $B$ is the zero matrix, that is: $AB=0$; ...
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1answer
23 views

Working on pivots intuition

I am learning linear algebra using Gilbert Strangs "Intro to LA" 4th edition. On problem set for chapter 2.3 "Elimination Using Matrices" I encountered a question I can't wrap my head around(problem ...
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1answer
23 views

find Direction vector

i got this problem ( very trivial I guess) $39x -51y =15$ $-52x + 68 = -20$ I've done the Gauss reduction and got this, Matrix: \begin{pmatrix} 1 & \frac{-17}{13} & \frac{5}{13} \\ 0 ...
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1answer
61 views

Is there an easier way to find the inverse of a 3x3 matrix?

I know the normal process is to do row operations to transform the matrix to get the identity matrix and then apply the same row operations in the identity matrix to get the inverse. But this process ...
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0answers
15 views

Fast method or direct generation of random upper triangular matrix using integer-restricted gauss elimination

I'd like to generate non-singular random upper triangular matrices of the following form: $$ \left[\begin{matrix} 1 & r_1 \cos{\alpha_1} & r_2 \cos{\alpha_2} & r_3 \cos{\alpha_3} & ...
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2answers
69 views

Why does the Gaussian-Jordan elimination works when finding the inverse matrix?

In order to find the inverse matrix $A^{-1}$, one can apply Gaussian-Jordan elimination to the augmented matrix $$(A \mid I)$$ to obtain $$(I \mid C),$$ where $C$ is indeed $A^{-1}$. However, I fail ...
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36 views

Improving a Numerically Instable 6th Order Polynomial Fit Using Gaussian Elimination

I had a requirement to implement polynomial curve fitting in software, which I have done using multiple regression and Gaussian Elimination with Partial Pivoting. I have done this for 2nd, 5th and ...
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21 views

Constrained System of Equations

$Ax=b$ is a linear system of equations with dimension of $n$ in which $A$ is real, symmetric, and positive definite (RSPD). The matrix $A$ can be also written as \begin{equation} A_{n \times n} = G_{n ...
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52 views

Does gaussian elimination always work?

If so, why don't we use that to get from any square matrix to a triangular matrix - from which can be deduced eigenvalues, determinant (product of eigenvalues) and diagonal matrix (since the diagonal ...
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1answer
24 views

How to know what steps to perform when doing Gauss-Jordan reduction?

I am currently studying linear algebra at university. We are currently doing Inverses of matrices using Gauss-Jordan Reduction, but no one has really explained how you are supposed to decide what ERR ...
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2answers
54 views

Linear Equation Problem Solving

I'm confused on how you can approach this problem using Gauss-Jordan elimination. I’m buying three chemicals, call them A, B and C. One kg (kilogram) of A takes up 100 cc (cubic centimeters) and ...
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1answer
75 views

Thomas Algorithm for Tridiagonal System

A professor gave us an assignment to solve a Tridiagonal system using Thomas Algorithm. Here is the exercise: I am lost as to what to do with that $(0.2\pi)^2$ and do I just calculate the ...
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1answer
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How to find the general solution to this matrix?

I am trying to find the general solution to this system of equations using an augmented matrix, and then using the gauss reduction technique, but i cant seem to get it into row echelon form no matter ...
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1answer
346 views

How to row reduce a matrix with complex entries?

I have been doing some practice questions for university, and one of them is regarding row reducing a complex matrix. From what I can work out, I think (i could very well be wrong) that the first ...
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5answers
58 views

Determine inverse matrix of $\left(\begin{matrix} 0 & 3 \\ 0 & 6 \end{matrix} \right)$ using Gauss-Jordan method

I need to find the inverse of the following matrix with Gauss-Jordan method, but apparently, checking with a calculator, it does not exist: $$\left(\begin{matrix} 0 & 3 \\ 0 & 6 \end{matrix} ...
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3answers
45 views

Is there any sufficient or necessary conditions for a matrix to do Gaussian Elimination?

For now I know that we can do Gaussian elimination to $ m\times n $ matrices. But is there any restrictions?
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44 views

gaussian elimination to solve a question (using a paramter)

I want to solve : x2+x3=0 -x1 -x3=0 x1-x2 =0 I got the $x_1 = -t, x_2=-t, x_3=t$. But the book has $x_1 = t, x2=t, x3=-t$. ...
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2answers
92 views

Gauss elimination applied to a 4x2 system of equations

I have used Gauss elimination just for a while, and I am still not comfortable, and most of the times I do mistakes and I do not arrive the the right upper triangular matrix. I have problem which ...
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2answers
75 views

Why this system is not linear and how that influences the number of solutions?

I have an exercise in my last assignment for linear algebra, it's the following: This system is not linear, in some sense $$ 4sin(\alpha) + 3tan(\beta) + 2cos(\gamma) = 1 \\ -sin(\alpha) + ...
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Apply gauss method and write elementary iteration matrices

I have a problem where I have to apply the Gauss method to 2 simple linear system of equations and write the elementary iteration matrices that I use for solving the systems. One of the system is: ...