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 to solve system of two equations with fractions?

I have the following system of two equations with two variables $0\leq{}x_c\leq{}1$ and $0\leq{}x_n\leq{}1$: ${\lambda{}}_c\left(-{\tau{}}_cx_c\frac{2C_c-r_cx_c}{C_c{\left(C_c-r_cx_c\right)}^2}+\frac{...
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what happens if i cant make a perfect triangle form gaussian elimination? and what does a determinant represent? [closed]

say I have a matrix of \begin{bmatrix}7&2&2&8\\0&6&8&13\\0&0&0&9\\0&0&0&10\end{bmatrix} would I just need to start over from the beginning until I have ...
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Proofs of conditional statements involving reduced row echelon form of a matrix. [closed]

Prove the following: Prove if an m x n matrix A has a column of zeroes, then the reduced row echelon form of A has a column of zeroes. If A has a row of zeroes, then the reduced row echelon form of ...
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Gaussian elimination with $4x-3y=11$ and $x-3y=-1$

(Meta: What's the notation for showing two equations on two lines within a set of square braces []?) Excuse syntax. I would like to use Gaussian elimination to solve the system $[4x+3y=11]$ and $[x-3y=...
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Conjecture with a three-diagonal system of equations

Everything in the sequel real-valued. And it is silently assumed that the denominators are non-zero, though the latter is not a trivial issue. Consider a piece of an (infinitely) large system of ...
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Transformation Matrix gaussian elimination of $ \left[T\right]_{B}^{B} $

So I have the question: Given vector space $M_2(\mathbb{R})$ with the linear transformation: $$ T(x)=AX-XA .\\ A=\left(\begin{array}{cc} 1 & 1\\ 1 & 1 \end{array}\right) $$ B is a base:$$ B=\...
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Least Squares via orthogonal projection and exact perp?

When you use least squares via orthogonal projection, is it common to not necessarily have an exactly 90 degree angle but close? I have attached an image. For example, I calculated a dot product ...
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A solution not quite adding up for Orthogonal Projection

The solution that is not adding up is in the link below. Specifically, the answer provided by Chad. I really enjoyed his answer but couldn't comment (not enough reputation) on one aspect that was ...
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Solving $AX=B$ for partially unknown $B$

I am trying to solve multiple systems of linear equations over $n$ variables $AX=B$, where $B$ is partially unknown. Let us assume that $A$ has shape $n' \times n$, $X$ has shape $n \times m$ and $B$ ...
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How should I choose a free variable in parametric form?

$$ \begin{pmatrix} 0 & 1 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 0 \\ \end{pmatrix} * x = \begin{pmatrix} 0 \\ 0 \\ 0 \\ \end{pmatrix} $$ What's the ...
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Trying to understand the Gaussian Transform in the $k^{th}$ reduction of a Matrix

The main role of the gaussian transform is to force the entries of a matrix $A\in\mathbb{R}^{n\times n}$ to be such that the $k^{th}$ entry of $A$ during the $k^{th}$ Gaussian reduction has all ...
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How to invert a family of matrices with parameter dependent dimensions and position dependent elements?

Let $\mu,\beta,\gamma,\kappa$ be real numbers (parameters) and let $n$ be an integer bigger equal one and let $\left(X_j\right)_{j=0}^{2 n} $ be a vector of real numbers. Consider a following ...
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Matrices: Using the row echelon [duplicate]

This is my system of linear equations $$ \left\{ \begin{array}{c} 1x_1+2x_2-1x_3=2 \\ -3x_1+1x_2-3x_3=1\\ 4x_1+ax_2-4x_3=b \end{array} \right. $$ My Rank matrix looks like this: $$ \begin{...
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What does it mean when the bottom row of a reduced row echelon form is all zeros with a 1 at the end?

I've just started my linear algebra course, and one question has me reduce an augmented matrix A of the form $A=\begin{bmatrix} n_{11} & n_{12} & n_{13} & a\\ n_{21} & n_{22} & n_{...
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The system should have only trivial solution

I want to calculate the sum $a+b+c+d$ such that the following system has only the trivial solution. $$\begin{pmatrix} 0& 5& -9& -c\\ 1&0&0&-d\\5&0&-a&0\\5&9&...
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Find rows of a matrix that are linearly dependent to some other rows

To avoid this being an XY problem, this is my background: I was reading theorem 2.2.2 of section 2.2.2 of this paper, which is about finding a relative interior point of a system $Ax \leq b$ ...
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Row independence and Column Independence

\begin{bmatrix} 3 & 4 & -1\\ -2 & 3 & 1\\ -9 & 5 & 4 \end{bmatrix} I tried to solve the matrix above using row reduction, I did the following steps. As soon as I found the ...
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Dixon/quadratic sieve factoring with Gaussian elimination

Given an integer $N$ to be factorized, seek $k+1$ integers $a_i$ such that the rest of $a_i^2$ divided with $N$ is a $k$-smooth integer $s_i$. The task is to multiply $s_{i_1}\cdots s_{i_m}$ such ...
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67 views

Finding all left inverse of a matrix B

Let $$ \mathbf{B} = \left[\begin{array}{cc} 0 & 1 \\ 1 & -3 \\ 0 & 0 \end{array}\right]. $$ My task at hand is to all left-inverse matrices of $\mathbf{B}$, without using the Moore-Penrose ...
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Rounding error in Gaussian elimination, partial pivoting

I would like to seek some clarification on how partial pivoting reduces the rounding errors in Gaussian elimination. For the ordinary Gaussian elimination I believe the rounding error arises from the ...
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methods to solve linear equations

are there any other methods to solve linear equations other than Gauss Elimination Method Gauss Jordan Method Matrix Inversion Method LU Decomposition Method Gauss Seidel iteration Method
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Solution of an overdetermined system of linear equations

According to my textbook "Matrix Operations for Engineers and Scientists - An Essential Guide in Linear Algebra" by the late Alan Jeffrey the following system of equations is impossible. To ...
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Eliminating Equations in a Block Matrix

I have the following linear system: $$ \begin{bmatrix} \mathbf{I} & \mathbf{0} & \mathbf{0} & -\mathbf{Z}^{T} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & -\mathbf{Y}^{T} \\ \...
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Why the number of linearly independent columns of a matrix doesn't change by if we apply Row operations

Can anyone please tell me why the number of linearly independent columns of a matrix doesn't change even if we apply row operations on the matrix? The column space does change by row operations but I ...
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How to find the accurate determinant of this matrix?

I'm solving the determinant of this matrix $\begin{bmatrix}1 & 2 & 3\\ 2& 2 &5\\ 3& 5& 1\end{bmatrix}$. I got two different answers by Gaussian Elimination. What's wrong with ...
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Intersection of two planes in $\mathbb{R}^4$

I have two planes which i have to intercept, but my answer isn't correct (i think) Plane I $= (-14, 8, 3, 3) + r(3,3−,3,0) + s(1, −1, −3, −1)$ Plane II $= (-7, 1, 3, 6) + u(-2, 0, -2, -2) + v(-3, 1, -...
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Why does Gaussian elimination for band matrices take $\omega$ operations to eliminate the first element?

I am currently reading through G. Strang’s “Linear Algebra and its applications” and there is this chapter about band and symmetric matrices which just does not stop confusing me. In this case we have ...
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Writing a vector a a sum of two vectors

Write $\mathbf u =\begin{bmatrix}9\\-1\\2\end{bmatrix}$ as a sum of two vectors where one is orthogonal to the plane $x+2y-2z=0$ and the other is parallel to the same plane. I've attempted to use ...
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How do I solve this system of four equations? Can it be done using Gaussian elimination?

I have the following system of four equations: $$\begin{align} &x_1 = \dfrac{1}{2} x_2 + \dfrac{1}{4} x_3 \\ &x_2 = \dfrac{1}{3} x_1 + \dfrac{1}{2} x_4 \\ &x_3 = \dfrac{2}{3} x_1 + \dfrac{...
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Rank of a matrix if one of the diagonal elements is $0$ during elementary row operations

I'm required to find the rank of a square matrix A of dimension $m * m$. While doing elementary row operations I encountered a $0$ at a diagonal position. I tried all the possible row exchanges, but ...
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Which of the following statements are correct for LU decomposition?

Which of the following statements are correct for LU decomposition? a. Columns of U are linear combinations of columns of A. b. Columns of U with pivot in them are candidate for basis set for column ...
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$Ax=0$ in a matrix

I am self-studying Linear Algebra, mainly through Khan Academy and some YouTube videos. I have encountered the following problem. Given $$A = \begin{bmatrix} 2 & -1 & -2\\ -4 & 2 & 4\...
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How would I convert this matrix to a system of equations?

Given a matrix $$A =\begin{pmatrix}1& 1& 1\\−2& 1& 3\\3& 2& 1\end{pmatrix}$$ use Gaussian elimination to compute the determinant $\det(A)$ of $A$ and to solve the system of ...
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Solving a linear equation system in $\mathbb{Z}_q$, where $q$ prime

I am wondering about the following: Assume you have a linear equation system of the form: $$Ax = b$$, where $A \in \mathbb{Z}_q^{n \times m}$, $x \in \mathbb{Z}_q^m$, $b \in \mathbb{Z}_q^n$ and we ...
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How to solve a system of 32 XOR equations via Gaussian Elimination?

In the beginning, I needed to solve a system of linear XOR equations. I was inspired by the answer to this question: how to solve system of linear equations of XOR operation? , and started solving my ...
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Column exchange still leaves the matrix unchanged

Professor Gilbert Strang writes in his notes ( does elimination on matrix $ U$ to reduce matrix into rref, denoted by matrix $R$ ) $$ ... U=\left[\begin{array}{llll} 1 & 2 & 2 & 2 \\ 0 &...
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Can we always write the elimination matrix $E$ as a lower triangular matrix?

Can we always write the elimination matrix $E$ as a lower triangular matrix? By elimination matrix I mean the matrix used in the Gauss elimination method.
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Is there any situation where the LDU decomposition is the same as the eigenvalue decomposition?

I was just wondering if there are any situation where the LDU decomposition is the same as eigenvalue decomposition (diagonalization)? The only way this can be possible if L and U are inverse so ...
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Using Gaussian Elimination to Reveal Secret Key

I've come across an article that shows how unknown information could be revealed from an ECDSA signature where there were some reuse of parameters use to calculate the signature. Specifically, the ...
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How to find matrix of linear map sample of vectors and their projection?

I would like to find matrix of linear transformation $f$, which projects vectors like this: $$f((x, y, z)^T)= (a, b, c, d, e)^T$$ How Can I find this matrix ? I am confused that vectors have different ...
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Row Echelon Form…Q

I can't see the mistake but it's probably really obvious... I have these equations to start: $$2x + y - z= 1$$ $$3x + 2y +z = 10$$ $$2x - y + 27 = 6$$ I was taught to work from left to right, worry ...
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Can two eigenvectors share one eigenvalue?

I apologize in advance for the mess. This is my attempt. The matrix in question is \begin{align} \textbf{A} = \frac{1}{100}\cdot\begin{pmatrix} 92 & 0 & -144 \\ 0 & 100 & 0 \\ -144 &...
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Solving system of equations, Gaussian Elimination

I am hoping to gain some insight on the characteristic differences between the following two distinct sets [(1) and (2)] of systems of equations. In particular, Gaussian elimination doesn't produce ...
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A Messy System of Linear Equations. (does anyone have a solver?)

Does anyone have a solver for the system of equations : $$ v_1=A+B- (c_5 a / c_1(c_8+ c_3c_5/c_1 )), $$ $$ v_2=Ae^{r_1 h }+Be^{r_2 h}- (c_5 a / c_1(c_8+ c_3c_5/c_1 )),$$ $$x_1=b-(1/c_1)(A(c_2+c_3/...
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How to prove that the first or a free column must be the linear combination of the previous columns?

For example, the matrix is $$ \left [\begin{matrix} 1 & 3 & -2 & 2 & 5 \\ 0 & 3 & -3 & 0 & 2 \\ -1 & 1 & -2 & -2 & 0 \\ 2 & 1 &...
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Rank of a matrix using colunm space

I'd like some verification of my solution method since I don't have any solution at hand. I'm given the following matrix and asked to determine the rank of it in function of $x$: $\begin{pmatrix} 2 &...
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How to solve a system of a single equation?

I’m trying to solve an apparently simple problem but couldn’t find a solution not involving advanced math. Let $10ax + (b-3a) = x - 3$ How do we solve equations of the given form? The solution of the ...
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Linear Algebra: Using Gaussian Elimination to obtain Row Echelon Form Matrix

I am learning Linear Algebra and I understand that we can use Gaussian Elimination to transform an augmented matrix into its Row Echelon Form using Elementary Row Operations. (ERO) One thing that is ...
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Symmetric matrics have pivots with the same signs as their eigenvalues

How would I prove the title statement? My Gilbert Strang Ed. 5 Linear Algebra textbook talks about $LDL^T$ factorization but I don't quite understand it. Online proofs also talk about Sylvester's Law ...
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Name of factors in Gaussian Elimination

Consider something like $\begin{pmatrix}1&2\\ 3& 4\end{pmatrix}^{-1}$. The classical way to do this is to write it next to an identity matrix, like the following: $$\begin{pmatrix}1&2\\ 3&...

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