Questions tagged [gaussian-elimination]
For questions on or related to the technique of Gaussian elimination, used in solving systems of linear equations.
682 questions
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A $4\times 4$ linear system with four parameters
Consider the following linear system
$$\begin{cases}
x + y + z + t = 1,\\
ax + by + cz + dt = 1,\\
a^2x + b^2y + c^2z + d^2t = 1,\\
a^3x + b^3y + c^3z + d^3t = 1\end{cases}$$
What is ...
2
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1
answer
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Finding All Left Inverses For a $3\times2$ Matrix
I’m trying to find all left inverses for this matrix A: \begin{bmatrix}1 & 3 \\2 & -4 \\3 & 4\end{bmatrix}
I know to start by row reducing [A⊤|I], which gives me: \begin{bmatrix}1 & 0 &...
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1
vote
1
answer
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Gauss-Jordan elimination for a $3\times 3$ matrix not matching given answer [closed]
I don't understand what I am doing wrong when attempting to get the inverse of this matrix using Gaussian Jordan Elimination.
$$
A = \begin{bmatrix}
5 & -5 & 5\\
1 & 4 & -4\\
-1 & -...
- 23
0
votes
2
answers
36
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Creating a matrix/system of equations to solve car distribution in three cities
A car rental company has offices in three cities: $A, B$ and $C$. Of the cars rented in $A$, $60\%$ are returned in $A$, $30\%$ are returned in $B$, and $10\%$ in C. Of the cars rented in $B$, $30\%$ ...
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1
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specifying an elimination matrix
This question comes from Gilbert Strang's Introduction to Linear Algebra.
Given matrix
$$
A = \begin{bmatrix}
1&1&0 \\
4&6&1 \\
-2&2&0
\end{bmatrix},
$$
create $3$ ...
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1
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0
answers
18
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Gaussian Elimination and How It Affects the Designation of Pivot Columns and Free/Basic Variables
I am learning linear algebra, and I am currently learning about Gaussian Elimination. I understand the method but have deeper, harder-to-articulate questions about Gaussian Elimination. Through my ...
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2
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Finding Jordan Normal Form of ridiculous matrix.
As an exercise for our exams we were tasked with finding the jordan normal form of the following matrix
where $\mathbb F_7$ denotes the finite field (modular arithemtic) over $\left\{0,1,2,3,4,5,6\...
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0
votes
1
answer
99
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Solving a System of Equations with Zero Determinant Matrix [closed]
I'm trying to learn FEA and I'm going through the first example problem from the "Practical Stress Analysis with Finite Elements" book by Bryan Mac Donald. Here is the problem: Matrix ...
2
votes
1
answer
61
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Finding value of unknown coefficient such that a linear system has a number of solutions
For which value/s of $a$ will the linear system have a unique, infinite, and no solution?
$\begin{cases} 2x+y+3z=1\\ x+2y+2z=2 \\ 3x+3y+az=3\end{cases}$
By Gaussian Elimination
$$
\left[
\begin{array}{...
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0
votes
0
answers
33
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Can I add a non integer multiple of a row to another row in gauss-elimination
Let's say I have the matrix.
\begin{bmatrix}
2 & 2 & 0 \\
1 & 1 & 1 \\
1 & 2 & 3
\end{bmatrix}
Can I perform the operation $r_2-\frac{1}{2}*r_1$ on this matrix? I know this ...
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2
votes
1
answer
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Continuous basis $(e_1(t),e_2(t))$
Assume that $t \mapsto A(t)$ is a $n \times n$ matrix-valued continuous function and that we know that $\ker A(t)$ has dimension $2$ for every real $t$.
Prove that there is are continuous vector-...
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1
answer
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Further explanation wanted on 'double Gaussian elimination' to triangularize a matrix.
I am trying to learn a more efficient way to triangularize a matrix. I found the following answer here on StackExchange which I found interesting, talking about 'double Gaussian elimination':
Short ...
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2
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2
answers
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Just a simple algebra question
So I have this question: Solve: x = 3y,
(a) x + y + z = 56,
(b) x - 2y - 3z = -25
So you can substitute 3y for x to then eliminate the z by multiplying (a) by -3z then solve accordingly and you get z =...
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votes
0
answers
36
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Effect of row operations on the sign of eigenvalues/ positive definiteness of remaining submatrix
Suppose, we have a real, symmetric, positive definite matrix $M$.
We know that M is positive definite if and only if all of its eigenvalues are positive.
My question is whether (Gaussian) eliminating ...
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Gaussian elimination choice for matrix
This questions isn't completely about mathematics, it is also part of computer science. I hope here is the correct place for it.
I studied about the gaussian elimination algorithm (I want to implement ...
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0
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1
answer
40
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How can I prove the following with Gauss's method? [closed]
$$\begin{vmatrix}
1 & 1 & 1 \\
\frac{1}{a} & \frac{1}{b} & \frac{1}{c} \\
bc & ac & ab
\end{vmatrix} =
\begin{vmatrix}
1 & 1 & 1 \\
a & b & c \\
b+c & a+c &...
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0
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0
answers
61
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What is the best algorithm to determine whether an $n \times n$ matrix is invertible or not? [duplicate]
I want to find the best algorithm to determine if an $n \times n$ matrix is invertible in high dimensions...
Is the best way to determine the invertibility of a matrix is to calculate the ...
0
votes
1
answer
91
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Updating a Matrix Determinant After Row Replacement
Given a square matrix A (of varying dimension), I am looking for an efficient algorithm or formula to recompute the determinant of that matrix if a row i is replaced with different values.
For example,...
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0
votes
1
answer
37
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Determinant at every step while finding matrix inverse
I've come to an intuitive conclusion that feels right and for which it seems there must be a proof, but I have been unable to locate one nor am I certain how to go about writing the proof. Therefore, ...
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2
votes
1
answer
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How to prove that this matrix is invertible by only elimination? Hoffman & Kunze exercise 1.6.12
Hoffman & Kunze exercise 1.6.12 wants a proof that this matrix is invertible
$$\begin{pmatrix}
1 & \frac{1}{2} & \frac{1}{3} & \dots & \frac{1}{n}
\\\\ \frac{1}{2} & \...
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1
vote
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 ...
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0
votes
1
answer
110
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Different eigenvalues using two different methods
I have to find the Eigen values of the following matrix:
$$
\begin{bmatrix}
2 & -1 & 0 & 0\\
0 & 3 & 0 & 0\\
0 & 0 & -2 & 0\\
0 & 0 & -1 &4
\end{bmatrix}...
0
votes
0
answers
40
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Question about inverse of gaussian transformation matrices (or atomic matrices in general)
I have a question about how the inverse of a gaussian transformation matrix, $M_k = I - m_k e_k^T$, is derived. The derivation I saw in a class is
\begin{align}
M_k^{-1} =& (I + \bar L)^{...
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1
vote
0
answers
88
views
Determinant of a Matrix using Gauss Elimination, inconsistent answers
I have worked through finding the determinant of the following Matrix
$$
\begin{pmatrix}
6 & -1 & 0 & 4 \\
3 & 3 & -2 & 0 \\
0 & 1 & 8 & 6 \\
2 & 3 & 0 &...
0
votes
0
answers
20
views
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
$$
A(...
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1
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Switching Rows in the Gauss Seidel method
So imagine I have been given the question as follows.
ax + by + cz = k [row 01]
dx + ey + fz = l [row 02]
gx + hy + iz = m [row 02]
Now if I solve this....the values converge to a certain value. But ...
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0
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1
answer
107
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Using Gaussian elimination for a parametric solution
On several occasions I've seen involving Gaussian elimination to solve a system of equations, while this method doesn't seem to add anyting to the process, and the system must be solved using regular ...
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2
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0
answers
45
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Rank of this matrix with a parameter: explanation about losing information
Consider the following matrix, where $k$ is a real parameter:
$$\begin{pmatrix} 1 & k & 1 \\ k & 1 & 1 \\ 1 & 1 & k \end{pmatrix}$$
I know I can study the zeroes of the ...
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427
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Solve the system of equations $x_1+10x_2-x_3=3,2x_1+3x_2+20x_3=7,10x_1-x_2+2x_3=4$ using the Gauss-Elimination with partial pivoting.
Solve the system of equations $$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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2
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1
answer
540
views
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 ...
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1
answer
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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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1
vote
1
answer
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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 &...
1
vote
0
answers
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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, ...
user1049492
1
vote
0
answers
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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-...
user1049492
4
votes
1
answer
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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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votes
1
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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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2
votes
1
answer
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views
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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0
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65
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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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vote
1
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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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3
votes
3
answers
98
views
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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1
vote
0
answers
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views
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 ...
1
vote
2
answers
427
views
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 ...
0
votes
0
answers
65
views
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, ...
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2
votes
1
answer
295
views
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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1
vote
0
answers
59
views
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 ...
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0
answers
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views
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 ...
1
vote
1
answer
169
views
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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