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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How can I prove the following with Gauss's method? [closed]

$$\begin{vmatrix} 1 & 1 & 1 \\ \frac{1}{a} & \frac{1}{b} & \frac{1}{c} \\ bc & ac & ab \end{vmatrix} = \begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ b+c & a+c &...
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What is the best algorithm to determine whether an $n \times n$ matrix is ​invertible or not? [duplicate]

I want to find the best algorithm to determine if an $n \times n$ matrix is ​​invertible in high dimensions... Is the best way to determine the invertibility of a matrix is ​​to calculate the ...
reza ibrah's user avatar
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Updating a Matrix Determinant After Row Replacement

Given a square matrix A (of varying dimension), I am looking for an efficient algorithm or formula to recompute the determinant of that matrix if a row i is replaced with different values. For example,...
Ood's user avatar
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Determinant at every step while finding matrix inverse

I've come to an intuitive conclusion that feels right and for which it seems there must be a proof, but I have been unable to locate one nor am I certain how to go about writing the proof. Therefore, ...
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How to prove that this matrix is invertible by only elimination? Hoffman & Kunze exercise 1.6.12

Hoffman & Kunze exercise 1.6.12 wants a proof that this matrix is invertible $$\begin{pmatrix} 1 & \frac{1}{2} & \frac{1}{3} & \dots & \frac{1}{n} \\\\ \frac{1}{2} & \...
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Gauss-Jordan elimination gives inconsistent matrix for a consistent system?

I am trying to get an analytical expression for a steady state of an ODE system governing a chemical reaction network via symbolic computer algebra systems. As an example for this question I'll take a ...
linkz's user avatar
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Different eigenvalues using two different methods

I have to find the Eigen values of the following matrix: $$ \begin{bmatrix} 2 & -1 & 0 & 0\\ 0 & 3 & 0 & 0\\ 0 & 0 & -2 & 0\\ 0 & 0 & -1 &4 \end{bmatrix}...
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Question about inverse of gaussian transformation matrices (or atomic matrices in general)

I have a question about how the inverse of a gaussian transformation matrix, $M_k = I - m_k e_k^T$, is derived. The derivation I saw in a class is \begin{align} M_k^{-1} =& (I + \bar L)^{...
TreeBark's user avatar
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Determinant of a Matrix using Gauss Elimination, inconsistent answers

I have worked through finding the determinant of the following Matrix $$ \begin{pmatrix} 6 & -1 & 0 & 4 \\ 3 & 3 & -2 & 0 \\ 0 & 1 & 8 & 6 \\ 2 & 3 & 0 &...
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Relationship between gauss elimination and vertex deficiency in associated graph

Currently reading through this document : https://www.jstor.org/stable/2100866 First few definitions (extracted from the paper) Given an undirected graph $G = (V,E)$ for each $v \in V$ we define $$ A(...
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Switching Rows in the Gauss Seidel method

So imagine I have been given the question as follows. ax + by + cz = k [row 01] dx + ey + fz = l [row 02] gx + hy + iz = m [row 02] Now if I solve this....the values converge to a certain value. But ...
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Using Gaussian elimination for a parametric solution

On several occasions I've seen involving Gaussian elimination to solve a system of equations, while this method doesn't seem to add anyting to the process, and the system must be solved using regular ...
mins's user avatar
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Do all matrices appearing in Gauss–Jordan elimination represent equivalent linear transformations?

There are three types of elementary row operations in Gauss–Jordan elimination: Swapping two rows, Multiplying a row by a nonzero number, Adding a multiple of one row to another row. From my ...
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Rank of this matrix with a parameter: explanation about losing information

Consider the following matrix, where $k$ is a real parameter: $$\begin{pmatrix} 1 & k & 1 \\ k & 1 & 1 \\ 1 & 1 & k \end{pmatrix}$$ I know I can study the zeroes of the ...
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Solve the system of equations $x_1+10x_2-x_3=3,2x_1+3x_2+20x_3=7,10x_1-x_2+2x_3=4$ using the Gauss-Elimination with partial pivoting.

Solve the system of equations $$x_1+10x_2-x_3=3,$$$$2x_1+3x_2+20x_3=7,$$$$10x_1-x_2+2x_3=4$$ using the Gauss-Elimination with partial pivoting. I tried solving the problem as follows: We have the ...
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LU decomposition of banded matrix with partial pivoting

Disclaimer: I'm rusty as can be in this department. I'm looking into how to implement a banded matrix LU decomposition with partial pivoting ($PA = LU$). So to start with I implemented regular matrix ...
Wout's user avatar
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What is the flaw in this Gaussian elimination?

$\newenvironment{sysmatrix}[1] {\left[\begin{array}{@{}#1@{}}} {\end{array}\right]}$ I was tasked with getting this to reduced row-echelon form: $$\begin{sysmatrix}{cccc|c} 1 & 3 & 1 &...
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Constrains of a free variables in a linear system?

We are given the task to find the numerical values of $a, b, c, d$ for the following equation $ax+by+cz+d=0$. We are given that this plane should intersect with the points $M=(4,4,4), N=(6,0,8), L=(5,...
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Can the steps used to solve for the Echelon form result in different answers, or will they always yield the same answer?

Starting Matrix: A= \begin{pmatrix}3 & -4 & 0 & 9\\ 2 & 4 &-1 & 0\\ 10 & 0 &-2 & -4 \end{pmatrix} Given = \begin{pmatrix}1 & 0 & -1/5& 0\\ 0 &...
Muhammad Owais Rafique's user avatar
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Showing that $\text{GL}_n(K)=BNB$

Let $G$=$\text{GL}_n(K) $ ($K$ is a field), $B$ is the subset of upper triangular matrices and $N$ is the subset of monomial matrices (or generalised permutation matrices). I am trying to show that ...
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Is the LU decomposition just Gauss-Jordan elimination?

I am watching Gilbert Strang's neat lecture on the LU decomposition, which is taught just after Gaussian elimination. $LU$ for a matrix $A$ was found doing $EA=U$ and finally $A=E^{-1}U$. Seems to me, ...
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Missing the point of LU factorization / decomposition

Gaussian Elimination The system of linear equations $Ax = b$ may be solved by using Gaussian Elimination (GE) arriving to a Row Echelon Form R of the augmented matrix $[A b]$, and then using back-...
Mah Neh's user avatar
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What can we get by row/column addition?

Let $\Bbb K$ be a field and let ${\bf M} \in {\Bbb K}^{n \times n}$ be a full rank matrix. Applying elementary row and column operations, one can transform $\bf M$ into the identity matrix. What can ...
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Why does Gaussian elimination sometimes work in rings where it should not?

I think it's best to illustrate this with an example. Take for instance the ring of integers modulo $6$. If I have the system of equations: $$ \begin{aligned} 2x + 2y &= 4 \\ 3x + 4y &= 3 \end{...
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Can I invert the hessian using row operations like this? [closed]

I put my derivations in this image here: [![enter image description here][1]][1] I am just using gaussian elimination by integrating all row elements. Is this acceptable? With this approach, we only ...
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Is there online calculator for calculating a row echelon form of a matrix over $\mathbb{Z}_n$?

There are many great online calculators for calculating row-echelon forms of matrices, such as this one. But now I need to calculate the row-echelon form of some large matrices over $\mathbb{Z}_n$ (...
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Can I invert a hessian matrix using gauss jordan elimination?

I don't understand why you can't invert a hessian matrix using gauss jordan method. Can't you integrate or differentiate an entire row (because they are linear operators) and then subtract/add/swap? ...
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Constrained Hankel matrix decomposition

I want to decompose a square Hankel matrix $\bf {H}$, whose elements below the anti-diagonal are zeros. The decomposed factors should necessarily meet the following constraints: $$\begin{equation} \bf ...
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Help with Gaussian elimination for a $3\times 3$ matrix

I'm currently working on a linear algebra problem and I'm having trouble understanding why my Gaussian elimination process is not yielding the same answer as the given solution. The matrix I'm working ...
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How to solve this system of linear equations using Gaussian elimination?

I'm having trouble with this problem from my linear algebra course. The problem is: A new restaurant owner decides to have 20 tables for her guests, a certain number of tables with space for 4 people,...
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Why the last row is a zero row when determining a basis for a subspace?

I have been around this a couple times, and I haven't been able to fully understand this. Happens that I solved a space basis problem in my linear algebra course but I (and apparently anyone I have ...
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Let $v_1 = (1, 0, 2), v_2 = (1, 1, a)$, and $v_3 = (a, 1, −1)$. Find the value(s) of $a$ for which $v_1, v_2$, and $v_3$ are linearly dependent

I'm struggling to solve this question without the use of the determinant (I'm not allowed to use it). I've tried setting up a matrix with the vectors and putting that matrix into reduced row echelon ...
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Scale factor of the determinant: I'm blind (SOLVED)

I have the following matrix $$A = \begin{pmatrix} 2 & 5 & 8 \\ 3 & 6 & 9 \\ 4 & 7 & 9 \end{pmatrix}$$ I already calculated the determinant with Laplace in two different ways, ...
Martin and Friends's user avatar
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Gauss elimination with constraints for variable's domain

Consider the following homogeneous, underdetermined linear system of equations: $$ \begin{align} c + d &= k + l \\ j + k &= c + d \\ i + l &= a + b \end{align} $$ As it stands (assuming e....
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Upper bounds for largest element in the kth column in Gaussian elimination

I am struggle with this exercise in numerical analysis topic. Let $A$ be an $n\times n$ nonsingular matrix and $A^{(k)}$ the matrix obtained in the $k$-th step of Gaussian elimination for $A$ with $A^{...
Oromion's user avatar
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Convert any non-singular square matrix to a strictly diagonally dominant one using only elementary row operations

Elementary row operations Swap the positions of two of the rows. Multiply one of the rows by a nonzero scalar. Add or subtract the scalar multiple of one row to another row. Strictly diagonally ...
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Gaussian Elimination when Order of Operations Matters

I've been self studying linear algebra and was hoping that someone could provide some insight and/or direct me to the right resource here. Let's say that we are playing the game lights out. Math.SE ...
Kelly C.'s user avatar
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How to solve nonlinear system of equations with Gaussian and Seidel?

For one of our college projects we’re required to solve numerous vector loops that get bigger and bigger. We weren’t quite taught Gaussian elimination and the Seidal iterative method but our lecturer ...
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Determine the base and the dimension of subspace $W$ given as generated space (set of linear combinations) of $3$-vectors in $\mathbb{R}^4$

Hello everybody I'm not certain with this question. So if lets say $$ W = L\bigl((1,1,0,-1),\, (0,-1,1,1),\, (3,1,2,-1)\bigr) \subset \mathbb{R}^4; $$ $L$ being the space generated or set of linear ...
Eternal Envy's user avatar
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Can you use row operations to reduce a matrix to either upper or lower triangular to find the eigenvalues?

For example, we have the matrix $$\begin{bmatrix}-1&0&1\\2&6&-14\\1&0&-1 \end{bmatrix}$$ Using row operations $R_3+R_1$ and $R_2 + 2R_1$ and finally $-1R_1$, we arrive at $$\...
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Diagonal dominance inequation

Let $LU$ decomposition for matrix $A \in \mathbb{R^{n \times n}}$ with diagonal dominance, computing by Gauss method. How to prove that $\frac{\displaystyle\max\limits_{ij} |u_{ij}|}{\displaystyle\max|...
LightM's user avatar
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Gaussian Elimination - where did I go wrong?

I have just learned about Gaussian Elimination and I decided to try an example question. I was trying to solve a question and I realised later that I had copied the question wrong but I still decided ...
Jeff's user avatar
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Graph theory beginning

We now consider different arithmetic operations for vectors and matrices and want to express the number of individual calculation steps as a function of n in the O -notation. By the number of ...
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Gaussian Elimination Elements $a^{(r)}_{ij}$

Let $A\in \mathbb{R}^{n\times n}$. We apply GE to it. Prove that: $\begin{align} a^{(r)}_{ij}&= a^{(r)}_{ij}=\frac{A\begin{pmatrix} 1 & 2 &\cdots & r & i \\ 1 & 2 &...
I Like Algebra's user avatar
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Growth of a completely pivoted Matrix

Let A be a CP matrix.( A completely pivoted matrix is one such that during the Gauss transformation with full pivoting there is no need to exchange rows or columns) We apply to it Gaussian elimination....
I Like Algebra's user avatar
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How can I do Gaussian elimination of a $32 \times 32$ bit matrix?

I have been looking at how to reverse the sigma operation in the sha256 hash and in several places I have seen that you have to make a $32 \times 32$ bit matrix and then solve it with Gaussian ...
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How to efficiently find this special case zero-eigenvector?

I have a real 3x3 symmetric matrix. I know that its 3 eigenvalues are 0 (within precision) and two real numbers >>0. What's the most efficient way to find the eigenvector corresponding to the ...
Jerry Guern's user avatar
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How does the Gaussian elimination method work to find the inverse matrix

How does augmenting a square matrix (LHS) with an identity matrix (RHS) and then reducing the square matrix to an identity matrix and performing the same operations on the identity matrix using ...
Ayse Kahraman's user avatar
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Formula for last element of zero row in symbolic gaussian elimination without swapping any rows

Suppose we have an overdetermined matrix with $n$ rows and $n-1$ columns. We augment it with a column-matrix and then do Gaussian elimination on the augmented matrix. Assume that no row swapping is ...
user2373145's user avatar
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Computing Inverse with Gaussian Elimination is not Backward Stable

Suppose that we are operating in a three-digit decimal floating point system. We want to use Gaussian Elimination with Partial Pivoting (GEPP) to find the inverse of a given invertible $2\times2$ ...
Hosein Rahnama's user avatar

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