Questions tagged [gaussian-elimination]

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

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Cannot solve linear system of equations

It would be nice if somebody could find my mistake for the following linear system of equations: $$ \left\{\begin{array}{rcrcrcr} -2x & - & 4y & - & z & = & -21 \\ -3x & + &...
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How do you decide which rows to swap for a matrix while doing gaussian elimination?

I'm trying to do LU factorization and for one of the examples I found online, the question is to solve the equation Ax=b with A and b as follows : A_and_b In their solution for finding the upper ...
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How do I solve the following system of equations by using the Gauss-Jordan method?

$$ \left\{\begin{array}{rcrcrcrcr} x & - & 2y & + & 3z & - & 4w & = & 10 \\ 2x & - & 3y & + & 4z & - & 5w & = & 18 \\ 3x & - & ...
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Particular matrix $A$ has infinitely many LU factorizations

Let $u = (u_1,\dots,u_n)^t \in \mathbb{R}^{n \times n}$ and $e_i$ the $i$-th canonical vector in $\mathbb{R}^{n \times n}$. Prove that for for $n \geq 2$, $A_n = u {e_n}^t$ has infinitely many LU ...
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Solve the system by RREF

Solve the system $$\left\lbrace \begin{array}{r@{}r@{}r@{}r@{}r} > x_1 & - x_2 & - 6 x_3 & = & -5 \\ >2 x_1 & - 2 x_2 & - 4 x_3 & = & -8 \\ >2 x_1 &...
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1answer
21 views

Property of symmetric positive definite related to diagonal entries and gaussian elimination

I am trying to prove the following property on symmetric positive definite matrices: Problem Let $A \in \mathbb{R}^{n\times n}$ be a symmetric positive definite matrix. Suppose that the gaussian ...
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Is there a simple proof that a non-invertible matrix reduces to give a zero row?

Let $A$ be a square matrix that is non-invertible. I was wondering if there is a simple proof that we can apply elementary row operations to get a zero row. (For a matrix $C$ to be invertible, I mean ...
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Alternating row and column operations for block Gaussian elimination to determine rank.

I am trying to determine the rank of a 6x6 symbolic matrix. The matrix can be represented as follows: $$ M = \begin{bmatrix} A_{ 3 \times 3} & R_{ 3 \times 3}A_{ 3 \times 3}\\ B_{ 3 \times 3} &...
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Potential problems with the Gaußian Elimination method?

We can use Gaußian Elimination (GE) to help us solve larger systems of equations. If we have a matrix $A \in \mathbb{K}^{m \times n}$, where $\mathbb{K} \in \{ \mathbb R, \mathbb C, \mathbb G \}$, ...
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Can I get this Gaussian elimination solution correct?

I have "Linear Algebra an introduction" ed 2 by A.O.Morris open in front of me at Exercise 1.2 (vi): "Find the reduced echelon matrix of: $\begin {pmatrix} 1 & 1 - \...
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How to solve $k_b \bar b - k_a \bar a + \frac{1}{t} \cdot k_a \bar a - \frac{1}{t}(\bar c - \bar a_1) = \bar a_1 - \bar b_1$

I need to solve $k_b$, $k_a$ and $t$ in this equation in a program I'm creating: $$k_b \bar b - k_a \bar a + \frac{1}{t} \cdot k_a \bar a - \frac{1}{t}(\bar c - \bar a_1) = \bar a_1 - \bar b_1$$ All ...
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Solving a system of equations with one variable in denominator

I have a system of equations in a program I'm writing. It's of the form: $$ \left\{ \begin{array}{c} \frac{a_1}{x}+b_1 y+c_1 z=d_1 \\ \frac{a_2}{x}+b_2 y+c_2 z=d_2 \\ \frac{a_3}{x}+b_3 y+c_3 z=d_3 \...
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1answer
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Assume $AX = C$. How to determine which entry of $BX - D$ is non-negative?

I posted this question on https://scicomp.stackexchange.com, but seems to receive no attention. As long as I get answer in one of them, I will inform in the other. Let $A,B$ be $n \times n$ matrices ...
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Linear Algebra - Gaussian Elimination Method on 3x4 Matrix

Solve the following homogeneous system using Gaussion elimination method. \begin{bmatrix} 1 & 3 & 5 & 1 & 0 \\ 4 & -7 & -3 & -1 & 0 \\ 3 & 2 & 7 & 8 & 0 ...
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Gaussian Elimination modulo 3 without fractions

Gaussian elimination can be used to solve a system of linear equations modulo a prime p. $p=2$ is trivial because no division is ever needed (only $\oplus$ operations). However, for $p=3$ it seems ...
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Are elementary row operators in linear algebra mutually exclusive?

There are three types of elementary row operations: I) row switching, II) row multiplication and III) row addition, corresponding to three kinds of row operation matrix. My question is that does ...
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Identification of reduced row echelon form and nuance in Gauss-Jordan elimination technique

I'm studying Linear Algebra: Step By Step by Kuldeep Singh, and on page 23 he shows this as an example of a matrix in reduced row echelon form: $$ \begin{pmatrix} 0 & 1 & 0 & 8 &...
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Given A=LU factorization, prove that the basis of column space A is the columns of L that correspond to the pivot columns of U

I understand that the basis of column space A is just the columns of A that correspond to the pivot columns of U. This is because U is just the reduced row echelon form. However, as mentioned in the ...
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Why use elimination method vs. substitution method in linear algebra?

What is the motivation to use the elimination method as opposed to the substitution method? I've always found the latter much easier. Are there any examples where the substitution method fails or when ...
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The necessary and sufficient condition of $rank(A)+rank(B)=rank(A+B)$ [duplicate]

Given two $m \times n$ matrices, find the necessary and sufficient condition of $rank(A)+rank(B)=rank(A+B)$. Here is my idea: Intuitively, I think it is like that A and B compliments each other. ...
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When should I use partial pivoting in Gaussian elimination?

Is there any criterion to decide whether to use naive Gaussian elimination or Gaussian elimination with partial pivoting? Why should I use partial pivoting if naive Gaussian elimination gives me the ...
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Solving problem with two invertible matrices using Gauss elimination method

So I solved this problem located in the picture below but I am not very confident I did it right so can someone look at it and see if it was solved right? Thank you! This phrase "Remark/Warning: ...
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Gauss-Seidel iteration method Convergence

I was hoping if someone can check my work on my proof on convergence of the Gauss-Seidel method. My friend and I are working on it and we have slightly different answer because of the addition and ...
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Proof of Gauss-Jordan elimination

I'm trying to prove that the Gauss-Jordan elimination works on any invertible matrix. Suppose that $A\in\mathcal{M}_{n}(\mathbb R)$ is an invertible matrix. We construct the matrix $B\in\mathcal{M}_{...
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Possible to gaussian eliminate matrix with a coefficient in front?

Is it possible to use Gaussian elimination on a matrix with a constant in front, like the one below? Do I multiply in 1/6 at the Reduced row echelon form stage? Or after parameterisation?
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For which $t\in\mathbb R$ the following system of linear equations can be solved

How do I determine for which $t \in \mathbb{R}$ the following system of linear equations can be solved, give the solutions matrix if there are any. $\begin{array}{lcl} 2x_1 & + & 4x_2 & +...
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Finding solution of $A\mathbf{x} = \mathbf{0}$ from rref over gf(2) [duplicate]

I'm working on the Gaussian elimination being implemented on gf(2). I have successfully reduced my 286*286 matrix into rref. Now I need to find the null space of this(please tell me how to do this ...
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29 views

For which Value of a Variable Gives Solutions of Linear System?

So I have found this question to be very vague and I am struggling to understand it and I believe I am close to the solution but I am not sure if it is correct. The question involves a linear system ...
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Help Finding Determinant with Row Reduction [closed]

I am trying to find the determinant of the matrix below using row reduction but keep getting different answers. My latest try resulted in a determinant of 384 yet, according to Symbolab it's -24. Can ...
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1answer
29 views

LU factorization of a singular matrix

I am trying to find the LU factorisation of the following matrix: $$A=\begin{pmatrix} 1 & 0 & 3 \\ 2 & 2 & 2 \\ 3 & 6 & -3 \end{pmatrix}.$$ Note that $A$ is singular. I ...
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1answer
60 views

What is the best way to solve square integer matrices of 8-bit?

Assume that we want to solve this linear system: $$A^TA x = b$$ Matrix $A$ is square and random integer of 8-bit, e.g numbers between 0 and 255. Vector $b$ is known as well and also integer of 8-bit....
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Matrix $E$ of the elimination

We have the matrix $$M=\begin{pmatrix}4 & 1 & 1 \\ 1 & 1 & 1 \\ 2 & 1 & -1\end{pmatrix}$$ I want to find the lower triangular matrix $E$ of the elimination. Is this matrix $E$...
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Formula for multiple integral equations

I need a formula to determine the value of $X$ & $Y$ in the below examples: \begin{cases} 26.95 - 26.95X - Y = 26.22 \\ 80.82 - 80.82X - Y = 79.23 \\ 53.87 - 53.87X - Y = 52.71 \end{cases} ...
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determinant of $4\times 4$ matrix by elimination

I am trying to find the determinant of this $4\times 4$ matrix. I got the wrong answer but I can't find the mistake The answer is supposed to be $-44$ but I got $-176$ the matrix $$ \begin{bmatrix} ...
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1answer
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Solving linear equation with spherical coordinates using gaussian elimination

I'm solving a linear equation to find a transformation (an homography) from (xi,yi) to (ui,vi) (i=1,2,3,4) : $$ui = \frac{c00*xi + c01*yi + c02}{c20*xi + c21*yi + c22}$$ $$vi = \frac{c10*xi + c11*yi ...
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How do you consider which element in the extended matrix is the pivot?

In fact, many articles said that the pivot must be the largest number in absolute value. But in wikipedia, the case was... $\begin{bmatrix} 0.00300 & 59.14 & 59.17\\ 5.291 & -6.130 & ...
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Solution for matrix equation with gaussian + column pivoting

\begin{array}{ccc|c} 7.000E+0 & 1.000E+0 & 1.000E+0 & 1.000E+1\\ 1.000E+1 & 1.000E+0 & 1.000E+0 & 1.300E+1\\ 1.000E+3 & 0.000E+0 & 1.000E+0 & 1.001E+3\\\...
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Why the null space will have dimension 2 given that the matrix is

Given that the matrix is \begin{bmatrix} 1 & -1 & 2 & 0\\ 0 & 1 & -1& 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ \end{bmatrix} thus ...
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Can I apply elementary row operation then find eigen values of a matrix?

Suppose if a matrix is given as $$ \begin{bmatrix} 4 & 6\\ 2 & 9 \end{bmatrix}$$ We have to find its eigenvalues and eigenvectors. Can we first apply elementary row operation . Then find ...
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Inverse using Gauss-Jordan elimination

Assuming $x,y,z \neq 0$, find the inverse of the following matrix using Gauss-Jordan elimination. $$\begin{bmatrix} 1&1&1&1\\ 1&1+x&1&1\\ 1&1&1+y&1\\ 1&1&1&...
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Help with Gaussian elimination

What is the memory, that is used with the cpu of the computer, to solving Gaussian elimination methods? I don't know how to answer this question. $\quad$,Thank you in advance
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Deleting a column from echelon form of a matrix is still in echelon form?

This is a true or false question which demands a counter-example in case it's false. The question says: If we delete any column of a matrix in echelon form, it will still remain in echleon form. ...
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How to solve Gaussian Elimination?

I have matrix solved as- 1 1 1 0 | 1 0 1 0 1 | 1 0 0 1 1 | 0 0 0 0 1 | 0 Now I can solve it as ...
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1answer
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Find the coordinates of the two lines where the distance between them is the shortest.

How can I use the minimum distance between two skew lines to find the exact coordinate where the distance between the two lines is the shortest? Let's say the lines have the equations: $$g = (1,3,5)^...
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Find the point where the two lines intersect.

I have two lines that intersect each other at a specific point. The equation of these lines is : $$g_1: x = b_1 + sr_1= \begin{bmatrix}1\\6\\1\end{bmatrix} + s\begin{bmatrix}2\\0\\1\end{bmatrix}, s \...
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1answer
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Linearly Dependent Columns of a Matrix

I want to make sure I'm understanding what the matrix of a linear transformation says about its null space and range. It's clear for me with rows (as this is how Gaussian elimination seems to be ...
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When does this matrix have no solultions, infinite solutions and 1 solution? And what are the solutions?

So im supposed to decide for what h and k this matrix has no solultions, infinite solutions and a unique solution $$\left[ \begin{array}{cc|c} 1&h&1\\ 3&3&k\\ \end{array} \right]$...
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1answer
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Unique polynomial in $C[x,y]$

I am working on the below problem: Let $I$ be the ideal generated by $x$ and $y$ in $\mathbb{C}[x,y]$. Prove that there is a unique (up to scalar) polynomial $f(x,y)$ of degree $2$ in $I^2$ which ...
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Can I always use Gaussian elimination to prove a matrix has no real eigenvalues?

If a matrix $M$ has no real eigenvalues, and if I don't know how to prove that its characteristic polynomial has no real roots, can I always prove it using Gaussian elimination on $M-xI$ and ...
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Showing a linear system has no solution

The system in question is $$\begin{cases} x_1 -x_2 + x_3 = -1 \\ -3x_1 +5x_2 + 3x_3 = 7 \\ 2x_1 -x_2 + 5x_3 = 4 \end{cases}$$ After writing this in matrix-form and performing row-operations we can ...

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