# 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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### Particular matrix $A$ has infinitely many LU factorizations

Let $u = (u_1,\dots,u_n)^t \in \mathbb{R}^{n \times n}$ and $e_i$ the $i$-th canonical vector in $\mathbb{R}^{n \times n}$. Prove that for for $n \geq 2$, $A_n = u {e_n}^t$ has infinitely many LU ...
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### Potential problems with the Gaußian Elimination method?

We can use Gaußian Elimination (GE) to help us solve larger systems of equations. If we have a matrix $A \in \mathbb{K}^{m \times n}$, where $\mathbb{K} \in \{ \mathbb R, \mathbb C, \mathbb G \}$, ...
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### Possible to gaussian eliminate matrix with a coefficient in front?

Is it possible to use Gaussian elimination on a matrix with a constant in front, like the one below? Do I multiply in 1/6 at the Reduced row echelon form stage? Or after parameterisation?
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### Solution for matrix equation with gaussian + column pivoting

\begin{array}{ccc|c} 7.000E+0 & 1.000E+0 & 1.000E+0 & 1.000E+1\\ 1.000E+1 & 1.000E+0 & 1.000E+0 & 1.300E+1\\ 1.000E+3 & 0.000E+0 & 1.000E+0 & 1.001E+3\\\...
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### Why the null space will have dimension 2 given that the matrix is

Given that the matrix is \begin{bmatrix} 1 & -1 & 2 & 0\\ 0 & 1 & -1& 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ \end{bmatrix} thus ...
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### Can I apply elementary row operation then find eigen values of a matrix?

Suppose if a matrix is given as $$\begin{bmatrix} 4 & 6\\ 2 & 9 \end{bmatrix}$$ We have to find its eigenvalues and eigenvectors. Can we first apply elementary row operation . Then find ...
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I have two lines that intersect each other at a specific point. The equation of these lines is : $$g_1: x = b_1 + sr_1= \begin{bmatrix}1\\6\\1\end{bmatrix} + s\begin{bmatrix}2\\0\\1\end{bmatrix}, s \... 1answer 41 views ### Linearly Dependent Columns of a Matrix I want to make sure I'm understanding what the matrix of a linear transformation says about its null space and range. It's clear for me with rows (as this is how Gaussian elimination seems to be ... 1answer 42 views ### When does this matrix have no solultions, infinite solutions and 1 solution? And what are the solutions? So im supposed to decide for what h and k this matrix has no solultions, infinite solutions and a unique solution$$\left[ \begin{array}{cc|c} 1&h&1\\ 3&3&k\\ \end{array} \right]$... 1answer 26 views ### Unique polynomial in$C[x,y]$I am working on the below problem: Let$I$be the ideal generated by$x$and$y$in$\mathbb{C}[x,y]$. Prove that there is a unique (up to scalar) polynomial$f(x,y)$of degree$2$in$I^2$which ... 1answer 53 views ### Can I always use Gaussian elimination to prove a matrix has no real eigenvalues? If a matrix$M$has no real eigenvalues, and if I don't know how to prove that its characteristic polynomial has no real roots, can I always prove it using Gaussian elimination on$M-xI\$ and ...
The system in question is $$\begin{cases} x_1 -x_2 + x_3 = -1 \\ -3x_1 +5x_2 + 3x_3 = 7 \\ 2x_1 -x_2 + 5x_3 = 4 \end{cases}$$ After writing this in matrix-form and performing row-operations we can ...