Questions tagged [gaussian-elimination]

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

For questions on or related to the technique of Gaussian elimination (also known as row reduction), used in solving systems of linear equations. Gaussian elimination is an algorithm for solving such systems. It is generally seen as a sequence of operations performed on the corresponding matrix of coefficients. These operations are:

  1. swapping two rows;
  2. multiplying a row by a non-zero number;
  3. adding a multiple of one row to another row.

Gaussian elimination can also be used to find the rank of a matrix, to compute the determinant of a matrix, and to determine the inverse of an invertible square matrix. It is named after Carl Friedrich Gauss (1777–1855).

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Can you use row and column operations interchangeably?

Is it possible to use row and column operations "at the same time" on a matrix $A$? So, for example, first subtracting $row_1$ from $row_2$, and then choosing to multiply $column_3$ by a constant $c$? Or do you have to "stick to one method" when…
Ius Klesar
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Why use Gauss Jordan Elimination instead of Gaussian Elimination, Differences

Why use Gaussian Elimination instead of Gauss Jordan Elimination and vice versa for solving systems of linear equations? What are the differences, benefits of each, etc.? I've just been solving linear equation systems, of the form Ax = B, by…
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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 each pivoting. From my understanding, in partial…
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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 to see why this actually works, and reading this…
rubik
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LU decomposition; do permutation matrices commute?

I have an assignment for my Numerical Methods class to write a function that finds the PA=LU decomposition for a given matrix A and returns P, L, and U. Nevermind the coding problems for a moment; there is a major mathematical problem I'm having…
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How do I show that a matrix is injective?

I need to determine whether this matrix is injective \begin{pmatrix} 2 & 0 & 4\\ 0 & 3 & 0\\ 1 & 7 & 2 \end{pmatrix} Using gaussian elimination, this is what I have done: \begin{pmatrix} 2 & 0 & 4 &|& 0\\ 0 & 3 & 0 &|& 0\\ 1 & 7 & 2 &|&…
eplosivemilton
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Time complexity of LU decomposition

I am trying to derive the LU decomposition time complexity for an $n \times n$ matrix. Eliminating the first column will require $n$ additions and $n$ multiplications for $n-1$ rows. Therefore, the number of operations for the first column is…
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Is there a way to know if a row reduction of a matrix has been done correctly?

I'm an undergrad taking the class of "Linear algebra 1". I came across a problem: sometimes we need to apply Gaussian elimination for matrices. Very quickly this skill is not much necessary as it's not a thinking skill but purely Technic. Yet,…
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What is an intuitive explanation to why elimination retains solution to the original system of equations?

I've studied linear algebra before, however, I wanted to come back to the foundations and understand it again from the beginning. I was looking the following inoffensive linear equations: $$ x - 2y = 1 $$ $$ 3x + 2y = 11 $$ and after elimination one…
Charlie Parker
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Solving systems of linear equations over a finite ring

I want to solve equations like this (mod $2^n$): $$\begin{array}{rcrcrcr} 3x&+&4y&+&13z&=&3&\pmod{16} \\ x&+&5y&+&3z&=&5&\pmod{16} \\ 4x&+&7y&+&11z&=&12&\pmod{16}\end{array}$$ Since we are working over a ring and not a field, Gaussian elimination…
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Decompose invertible matrix $A$ as $A = LPU$. (Artin, Chapter 2, Exercise M.11)

Decompose matrix $A$ as $A = LPU$, where $A \in Gl_n( \mathbb{R}^n)$, $L$ is lower triangular, $U$ is upper triangular with diagonal elements of $1$, and $P$ is a permutation matrix. It is fairly easy to decompose any invertible such $A$ as $PA =…
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How to reduce matrix into row echelon form in NumPy?

I'm working on a linear algebra homework for a data science class. I'm suppose to make this matrix into row echelon form but I'm stuck. Here's the current output I would like to get rid of -0.75, 0.777, and 1.333 in A[2,0], A[3,0], and A[3,1]…
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Understanding matrices as linear transformations & Relationship with Gaussian Elimination and Bézout's Identity

I am currently taking a intro course to abstract algebra and am revisiting ideas from linear algebra so that I can better understand examples. When i was in undergraduate learning L.A., I thought of matrix manipulations as ways of solving $n \times…
still_learning
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How can I use Gauss elimination to solve equations with Modular arithmetics?

I've given some equations look like this. $a_{1,1} x_1 + a_{1,2} x_2 + a_{1,3} x_3 + ... + a_{1,n} x_n\equiv 1 \mod p$ $a_{2,1} x_1 + a_{2,2} x_2 + a_{2,3} x_3 + ... + a_{2,n} x_n\equiv 1\mod p$ $...$ $a_{m,1} x_1 + a_{m,2} x_2 + a_{m,3} x_3 + ...…
Love Paper
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Algebraically-nice general solution for last step of Gaussian elimination to Smith Normal Form?

My question takes a little bit of preamble: it concerns a well-known and solved problem, but I am looking for a solution with a particularly nice property. $\newcommand{\matrix}[4]{\left( \begin{array}{cc} #1 & #2 \\ #3 & #4 \end{array} \right)} …
Peter LeFanu Lumsdaine
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