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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Is vector in the span?

So, from my understanding, a span is simply the space that can be made up of specified vectors. So if I wanted to find if vector $(1,2,3,4,5)$ was in the span of some vectors, I assume I should use ...
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Determine rank of $ A = \begin{bmatrix} 2 & 1 & -2 & 1 \\ 4 & 1 & -2 & -3 \\ 1 & -1 & 2 & -3 \\ 2 & 2 & -4 & -5 \\ 3 & 1 & -2 & 2 \end{bmatrix}$

Could you give me your feedback ? I've verified with https://matrix.reshish.com/rankCalculation.php but maybe there are things that could be done differently Determine the rank of the following ...
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Traffic density of road network using System of Equations, Matrices and Gaussian Elimination

Could you give me your feedback ? We consider the following road system with the four intersections $A, B, C, D$. The numbers and variables correspond to the traffic density (Number of vehicles per ...
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Under which conditions is the following linear equation system solvable $x_1 + 2x_2 - 3x_3 = a$ $3x_1 - x_2 + 2x_3 = b$ $x_1 - 5x_2 + 8x_3 = c$

I'm unsure about the problem below Under which conditions is the following linear equation system solvable ? $$x_1 + 2x_2 - 3x_3 = a$$ $$3x_1 - x_2 + 2x_3 = b$$ $$x_1 - 5x_2 + 8x_3 = c$$ We set up ...
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Determine inverse matrix of $A = \begin{bmatrix} 1 & 2 & -1 \\ -3 & 1 & 2 \\ -2 & 2 & 1 \end{bmatrix}$

Could you give me your feedback ? Determine the inverse of the following Matrix: $$A = \begin{bmatrix} 1 & 2 & -1 \\ -3 & 1 & 2 \\ -2 & 2 & 1 \end{bmatrix}$$ We want to ...
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How many of each specific coins are in a bottle?

Question Suppose you have a bottle that contains exactly twenty-two U.S. coins.These coins only consist of nickels(\$0.05), dimes(\$0.10), and quarters(\$0.25). In addition to the types of coins, you ...
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Numerical Analysis Gauss Elimination Question

I'm trying to solve this question but I'm stuck. Can you help me? Find approximate solutions of the following system using Gauss Elimination with partial pivoting. Use 4 significant digits in all ...
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How difficult is Gaussian elimination with automorphism?

Recently, I've had an exchange with someone about linearizing a randomly selected quasigroup. Let's say, this quasigroup is evaluated as $q(x,y) = Ax + By +c$ where $q$ is the quasigroup operation, $+$...
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Linear system - I do something wrong, I don't know what

$$ A := \begin{pmatrix} 5 & 0 & 0 & 0 \\ 0 & 2 & 2 & 1 \\ 0 & 0 & -4 & 0 \\ 0 & 2a & 0 & -2 \end{pmatrix} $$ Find $\mathcal{L}=(A,0), a \in \mathbb{R}$...
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Do the rows used in row operations during LU factorisation matter?

A method I have seen for finding the LU factorisation of a matrix is that U is the row echelon form of A. The row operations we perform on A to get to U must involve replacing $R_i$ by $R_i - kR_j$ ...
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Complete (?) list of row reduction applications

I continue to be amazed that row reduction is still, by far, the single most useful technique I've learned for solving linear algebra problems. Even as I do practice problems for my qualifying exam, ...
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I have a general solution as an affine map in matrix form. How can I tell if its surjective and/or injective?

So for example, the general solution is (1,2,3)d + (4,5,6) = (a,b,c) How could you tell if this is surjective and/or injective?
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I need help find the basis of a subspace in $R^4$ for three vectors

Find a basis for the subspace of $\mathbf{R}^4 $ spanned by the given vectors (1,1,-3,-2), (3,0,3,-3), (3,-1,-2,13) A problem that was similar involved reducing an augmented Matrix to RREF but that ...
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How to express the span and basis in terms of a vector space

I'm trying to solve the following question from Hefferon's Linear Algebra, but I cannot match the solution manual: The question is: Find the span of each set and then find a basis We're given this set:...
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Smith normal form of a real symmetric matrix

I have a real integer symmetric matrix $A$ for which I know has eigenvalue decomposition $$A=QDQ^T$$ I know that $D$ is a vector of integers, but $Q$ is an orthogonal matrix consisting of real, ...
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How do you go about Gaussian elimination modulo p?

The problem at hand is specifically to find the inverse of a matrix using Gaussian elimination modulo 29. I am familiar with the process of regular Gaussian elimination and modular arithmetic but not ...
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Gaussian Elimination and Eigenvectors

I've read in many places that Gaussian Elimination cannot be used to find the eigenvectors of a matrix. I don't understand why. Assume we have the matrix $\mathbf{A}$ and we know the eigenvalues $\...
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Applications of the Gauss-Jordan elimination method

Aware of my minimal knowledge of linear algebra, I was seeing the power of the Gauss-Jordan elimination method. In particular, it can be used for: solve the linear systems $A\,X = b$; calculate the ...
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Adding two terms in a Simultaneous Equation.

Sorry, but I have a problem where I must add two terms like this in a simultaneous equation: 2x + 4y = 32 2x - 3y = 11 I want to add the terms ...
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Problems with Simultaneous Equations

I have a problem with: 2x + 4y = 32 2x - 3y = 11 I know that I should add because the signs are different. But if I tried to eliminate the ...
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4 votes
3 answers
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Inverse of a 2x2 matrix with an example

Need to find inverse of this matrix: $ \begin {bmatrix} 1 & 3/5\\ 0 & 1\\ \end {bmatrix} $ This is how it has been solved: $ \begin {bmatrix} 1 & 3/5\\ 0 & 1\\ \end {bmatrix} $ $ \...
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Does a (square) full rank matrix have identical column and row spaces?

Say we have a full rank matrix A (i.e., the rows are linearly independent and the same is true for columns). Since using the basic procedures of swapping, scalar multiplication, and addition, we can ...
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Algorithm to find inverse of matrix

I've been stuck in this problem for a while now. This is what the solution is supposed to be. Any ideas?
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Can a non-homogenous augmented matrix be converted to homogenous and then be solved using Gaussian elimination?

Pretty much the title. I want to know if we can convert a non-homogenous linear equation into a homogenous one. I'm specifically asking about the equations that can be turned into an augmented matrix (...
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-1 votes
1 answer
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Solving system of $2$ linear equations in $3$ unknowns using Gaussian elimination [closed]

Using Gaussian elimination, solve the following system of equations. $$ \begin{aligned} 0.6 x + 0.3 y - 0.4 z &= -1.9\\ -4.6 x + 0.5 y + 1.2 z &= -1.3 \end{aligned} $$ I am unable to solve it....
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Gauss elimination over $\mathbb{Z}_2$

I should solve this system of equations over $\mathbb{Z}_2$: $3x_1 + x_2 + 2x_3 + x_4 = 2$ $-x_1 + x_2 + x_3 - x_4 = 1$ $-5x_1 + x_2 + 3x_3 + 3x_4 = 1$ So I tried to set up the matrix, which would be $...
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Linear Algebra Question: Row Reduction and Echelon Form

I am just starting out in learning Linear Algebra and I have been trying to figure out something that has me confused. Per the rules, as I understand them, when reducing a row to obtain a matrices Row ...
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Gaussian elimination: all variables in row are zero but constant is non-zero?

I'm starting linear algebra next week and been trying to get a little ahead. I'm working with a practice sheet I found through Google. One of the problems has the equations $2x+5y=9$ $x+2y-z=3$ $-3x-...
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Tricky question about... a infinite reduced row echelon matrix

Let's say I want to write the polynomial function $p(x)$ that passes the points $(1,1!)$, $(2,2!)$, $\ldots$, $(n,n!)$, that will be of the form $p(x) = a_0 + a_1x + a_2x^2 + \ldots + a_nx^n$. What I ...
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4 votes
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Growth factor Gaussian elimination

I'm looking at the exercises in the book "Numerical Linear Algebra" by Trefethen and Bau. I'm quite stuck at question 22.2: Experiment with solving $60\times60$ systems of equations $Ax=b$ ...
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The essence of Gaussian elimination

I'm confused about the essence of Gaussian elimination. Suppose there is a linear map between V and W. $T: V\rightarrow W$. Then there can be a matrix associated with the linear map $T$. Then apply ...
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Solving System of Equations using Gauss/Gauss-Jordan (Matrix)

System of equations $$\begin{align*} x -2y +3z &= 2\\ 2x -3y +8z &= 7\\ 3x -4y +13z &= 8 \end{align*}$$ In a augmented matrix, $3\times 4$ $$\left(\begin{array}{crc|c} ...
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System of linear equations to solve formula for integral

Determine the coefficients a,b and c so that the formula is valid for all polynomails of degree 5. $\int_{0}^{1} f(x)dx=\frac{1}{90}[af(0)+bf(0.25)+cf(0.5)+bf(0.75)+af(1)]$ This question is in my ...
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Row reducing an integer matrix

Given a $n\times n$ integer matrix, what is the best row reduction that can be found using only integer row operations of the form: an integer multiple of row $i$ can be added to row $j$ row i can be ...
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How where to practice solving systems of equations over finite fields

so in college I am taking Linear Algebra and we're having a test. In this test we have to solve a system of equations over a finite field. Let's have a finite field $Z_n = \{0, 1, ..., n - 1\}$. So ...
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4 answers
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Show that $\sum_{2}^{n} (k^2-k)=\frac{n^3-n}{3}$

I am reading about Gaussian elimination, and the book gives the equation for the number of multiplications/divisions needed to solve a nXn matrix $$\sum_{2}^{n} (k^2-k)$$ The formula I can understand, ...
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Number of multiplications needed for Gaussian elimination on a kXk matrix

I am reading about LR-factorisation and the book is demonstrating the number of multiplications different methods of solving matrix equations will requre. $A^n\vec{y}=\vec{x}$ where A and $\vec{x}$ ...
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Inverse of lower triangular matrix in terms of column vectors

Let $A$ be an $n \times n$ lower triangular matrix given with all diagonal entries equal to 1: $$ A = \left[ \begin{matrix} 1 & 0 & \cdots & 0 & 0 \\ a_{2,1} & 1 & \...
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1 vote
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Do pivots have to be consecutive across the columns or can their be a break?

I have a 3x3 matrix as described below. Is the rank of this matrix 1 or 2? Or otherwise asked, is it okay to have "gaps" in the pivots across the columns? $$ \left(\begin{array}{cc} 1 & ...
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1 answer
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Integer row reduction without scalar multiplication

For which matrices is it possible to find the (unreduced, and with arbitrary pivot) Echelon form of a matrix following Gaussian elimination, but only with the row operations: Adding/subtracting one ...
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Automorphism of $k[x_1,\ldots,x_n]$ [duplicate]

Say we have $r$ linear polynomials $p_1,\ldots,p_r\in k[x_1,\ldots,x_n]$ which are linearly independent. How can we establish an automorphism of $k[x_1,\ldots,x_n]$ such that each $p_i$ goes to $x_i$? ...
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5 votes
1 answer
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Do the left and right "parts" of the matrix used when performing Gaussian Elimination have names?

Given a matrix used to perform Gaussian Elimination like this: $$ \begin{bmatrix} 1 & -3 & | & 1 \\ 2 & -7 & | & 3 \end{bmatrix} $$ Which would be derived from a system of ...
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Show that two combinations of two vectors in two dimensional space takes 7 multiplication

I was reading the book Linear Algebra and Its Applications by Gilbert Strong(4th edition page-15) and in the first chapter came across the line Two combinations of two vectors in two dimensional ...
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Does RREF affect the augmented column?

If I have a system of equations I want to solve by row reducing a matrix, do the standard RREF rules also apply to the augmented column of constants? Say we have an inconsistent system with a bottom ...
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1 answer
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Creating zero values in a matrix without following typical Gauss - Jordan steps.

I was presented with the following Matrix in an assignment and asked to find the value for B that would make the rank of the matrix 2. \begin{pmatrix}3&-3&0\\ 3&-1&2\\ b&0&2\...
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If $a_{ij}$ typically refers to row i and j column of a matrix, then why does the notation for interchanging rows refer to j as a row?

I am confused by the notations I am seeing in my linear algebra class. Here is the book I am using and page I am on. $R_i \leftrightarrow R_j$ means to interchange rows $i$ and $j$ of a matrix. But ...
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Please show steps using Gaussian Elimination to solve X for a set of modular linear equations

There is a linear system of 4 equations in field $Z_p$ with 4 unknowns $k_1$, $k_2$, $x_1$, $x_2$. The other variables s, r and h are known. Please show me all the steps using Gaussian Elimination or ...
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0 votes
1 answer
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A symmetric matrix producing a smaller symmetric matrix upon Gaussian elimination

Given a real $n \times n$ symmetric matrix $A$ and $a_{11}$ is non-zero, if you use Gaussian Elimination to reduce it to $$ \begin{pmatrix} a_{11} & a_1^T \\ 0 & A_2 \end{pmatrix} , $$ $A_2$ ...
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1 vote
1 answer
72 views

Gauss-Jordon Elimination with nxm matrix

Is it okay to use Gauss-Jordan elimination on a vertically-rectangular matrix (e.g. $3\times2$)? (Note: I understand about reduced row echelon form for a horizontally rectangular matrix such as a $2\...
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3 votes
2 answers
162 views

Given matrix $X$, how to find elementary matrices $E_1$, $E_2$ and $E_3$ such that $X = E_1 E_2 E_3$?

Given $$X = \begin{bmatrix} 0 & 1\\ -2 & -18\end{bmatrix}$$ find elementary matrices $E_1$, $E_2$ and $E_3$ such that $X = E_1 E_2 E_3$. My attempt I did 3 row operations from $X$ to get to $...
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