0
votes
0answers
34 views

Solve linear equations [on hold]

\begin{bmatrix} 0 & 0& 1& 1& 1&0 \\ 0 & 0& 0& 0& 2&1 \\ -3 & 0 & 0 & 0 & -2& 0\\ 4& 4& 0& 0& 0 & -1\\ ...
0
votes
1answer
27 views

Transformation of inverse to a system of linear equations

I have $X = (U'WU)^{-1}U'$ to be solved. Suppose $U'$ is $3 \times 7, W$ is $7 \times 7$ positive definite matrix, $U'$ is of rank 3. So, I transformed $(U'WU)^{-1}U'$ as $(U'WU)^{-1}U'WU = I\\ XWU ...
0
votes
0answers
38 views

Intersection of lines in 3D space [closed]

Given two or more pairs of points in 3D space, I should calculate the intersection of the lines passing through each of these pairs of points: for each pair of points, I get the linear system of two ...
1
vote
0answers
17 views

Verification - Matrices and Linear Equations Part 1

Would just like to verify my Answers and bounce off my ideas and thinking with someone as I feel quite alone in this course. I am usually great at maths and enjoy it but these matrices and linear ...
1
vote
1answer
29 views

Solution to two equations with three unknowns

So I'm a student studying through correspondence and I need some help. This is an assignment question, and I have tried everything I know how, to answer it which has lead me to the conclusion that ...
1
vote
2answers
223 views

Matrix Equation- solution

Sir, We have given $A= \begin{bmatrix}q_1 & q_2&q_3 \\ q_4 & q_5&q_6\\ q_7 & q_8&q_9 \end{bmatrix} \tag 1$. A is a matrix with determinant 1,orthogonal , invertible and ...
2
votes
0answers
21 views

Find a reduced echelon basis from a reduced echelon matrix.

The reduced row matrix was this ---> $\begin{pmatrix}1&2&0&1&0\\0&0&1&3&0\\0&0&0&0&1\\0&0&0&0&0&\end{pmatrix} = 0$ So i computed ...
0
votes
3answers
33 views

Prove that $b^2 pr =q^2 ac$ using matrices

Let $i_1,i_2$ and $j_1,j_2$ be non-zero real roots of $ax^2+bx+c$ and $px^2+qx+r$ respectively, where a,p $\neq$0. If the system of equations $ i_1y+i_2z=0$ and $j_1y+j_2z=0$ has a non-trivial ...
1
vote
1answer
36 views

Solving an augmented coefficient matrix so there are infinitely solutions

I am trying to figure out this math problem. For what values $a,b$ does the linear system have infinitely many solutions? This is the matrix $$ \left[ \begin{array}{ccc|c} ...
1
vote
2answers
53 views

For which values of $a, b$ does the system of equations NOT have any solutions?

I am trying to solve this math problem: For which values of $a$ and $b$ does the linear system represented by the augmented matrix not have any solution? $$ \left[ \begin{array}{ccc|c} ...
0
votes
1answer
25 views

Cramer's rule and linear dependence/independence test

When you have the system of equations: $$ax + by = e\\cx + dy = f$$ And you do some row operations to eliminate $y$, we get: $$x = \frac{ed-bf}{ad-bc}\tag{1}$$ If we eliminate $x$ we get: $$y = ...
0
votes
1answer
29 views

Gaussian elimination easily?

Let $$\left( {\matrix{ 1 & 0 & { - 3} \cr 0 & 2 & {\lambda + 3} \cr 0 & 0 & {5 - {\lambda ^2}/2 - 3\lambda /2} \cr } \left| {\matrix{ { - 3} \cr 2 ...
5
votes
1answer
97 views

When is the equation $Ax = b$ solvable in the integers?

Let $A$ be an $m\times n$ matrix with integer entries, $b$ a column-vector with $m$ integer entries. Suppose the equation $Ax = b$ has infinitely many solutions. It is clear that the general ...
1
vote
1answer
50 views

An example on my book that asks for the basis of an eigenspace

Let $$ A = \begin{bmatrix}4&-1&6\\2&1&6\\2&-1&7\end{bmatrix}$$ An eigenvalue of A is 2. Find a basis for the corresponding eigenspace. Solution: Form $$A-2I = ...
0
votes
3answers
48 views

How does a row of zeros make a free variable in linear systems of equations?

I don't understand how a row of zeros gives a free variable when solving systems of linear equations. Here's an example matrix and let us say that we're trying to solve Ax=0: $$\left[ ...
1
vote
2answers
24 views

Get variables with Matrix

I try to get the variables for this equation: $$\begin{cases} 6x_1 + 4x_2 + 8x_3 + 17x_4 &= -20\\ 3x_1 + 2x_2 + 5x_3 + 8x_4 &= -8\\ 3x_1 + 2x_2 + 7x_3 + 7x_4 &= -4\\ 0x_1 + 0x_2 + 2x_3 ...
2
votes
0answers
47 views

How to solve this system of inhomogeneous differential equations

In some past exam papers for the Maths course that I attend,I found this example and I would really appreciate if someone looked at my solution. It goes like this: Find general solution to $$ y_1' = ...
2
votes
2answers
65 views

Solve a system of linear equations

$\newcommand{\Sp}{\phantom{0}}$There is a system of linear equations: \begin{alignat*}{4} &x - &&y - 2&&z = &&1, \\ 2&x + 3&&y - &&z =-&&2. ...
0
votes
2answers
37 views

Method for Finding Matrix-Inverse Through Gauss-Jordan?

When trying to find the inverse of the n$\times$n matrix $A$, one way of going about it is by solving $AX=I$, wherein $I$ is the n$\times$n identity matrix, and $X$ is some n$\times$n matrix which is ...
0
votes
1answer
17 views

Matrices in systems of linear equations

I've been working on matrices lately. Currently, I am stuck on solving systems of linear equations using matrices. I've read the following article which has proved very helpful in understanding the ...
1
vote
1answer
72 views

Eigenvalues and Eigenvectors for matrix. Complex Eigenvalues

How can I find out the eigenvectors for this matrix: $$A= \begin{pmatrix} -3 &0&0\\ 0&3&-2\\ 0&1&1 \end{pmatrix} $$ I found the eigenvalues: $\lambda_{1}=-3$, ...
1
vote
1answer
41 views

how to find matrix A from complete solution to Ax=b

I am trying to solve a problem. I was stuck.Any help is appreciated. The complete solution to $Ax=\left[\begin{array}[c]{rr}1 \\3 \end{array}\right]$ is $ x= ...
2
votes
1answer
56 views

Is it possible for a system of equations to have a non-zero determinant and no solution at the same time?

I am quite confused by the solution I was given for the following problems: a) Solve the following system of equations using Gauss elimination only: $2x - y = 5$ $-x + 2y = -4$ $3x - y = -1$ b) ...
0
votes
1answer
34 views

Difficult augmented matrices question.

I'm currently revising for a maths module that I am taking as part of my physics degree. The final part of the matrices section of a paper I was doing included this question: Solve the this set of ...
0
votes
0answers
16 views

Criterion of removal of equations from overdetermined system

Consider the problem of solving overdetermined system Ax = b; In the problem I am trying to solve (from the field of spectral unmixing) number of unknowns usually varies between N = 2 and 5 and the ...
0
votes
1answer
54 views

Using QR decomposition to solve a system of equations with a singular matrix

If $A\in\mathbb{R}^{n\times n}$ is singular and $x,b\in\mathbb{R}^{n}$ are such that $Ax=b$, am I right in thinking that the upper triangular matrix $R$ of $A$'s $QR$ decomposition must have at least ...
0
votes
3answers
51 views

Choose h and k such that the system has a solution, a unique solution and many solutions.

Im learning linear algebra, and im tasked with choosing $h$ and $k$ such that this system: $$ \begin{cases} x_1+hx_2=2\\ 4x_1+8x_2=k\\ \end{cases} $$ Has (a) no solution, (b) a unique solution, and ...
0
votes
1answer
63 views

Solve Coupled System of Equations via Matrix

I have a coupled system of three equations that I am trying to solve via matrices and I am having trouble figuring out how to write out my matrices. My three equations are as follows: $-sx+sy=0$ ...
0
votes
1answer
32 views

Can someone please provide an intuition behind cramer's rule?

See question. I usually get concepts like this very quickly (no studying required), but this one looks like Chinese. Can someone please help me understand a brief intuition behind Cramer's rule for ...
0
votes
1answer
29 views

Methods for solving linear systems

This is such a basic topic but there are so many different methods proposed for solving a linear system of equations. I recently found a very good source but couldn't really make sense of all the ...
2
votes
2answers
43 views

Determining the necessary values for a matrix' coefficients to achieve a certain rank.

I'm having a headache with this... Given the augmented linear system matrix: $$A = \begin {cases} 1 & 0 & 0 & 2 \\ 0 & a-2 & 0 & 0 \\ 0 & 0 & b + 1 & c \\ 0 ...
1
vote
2answers
250 views

Real world situation with System of Equation with 3 variables?

Where do you run into a real world situation involving 3 variables and 3 equations? Can someone think of a specific example from business, etc? I recall taking an operations research course that ...
0
votes
0answers
26 views

Find angle-preserving transformation matrix given 2 points

I asked a similar question yesterday about finding an affine transform matrix given the same 2 points in both coordinate systems. I was told that there was only a unique solution, if the scaling was ...
0
votes
1answer
21 views

Complexity of sparse back substitution

What is the complexity of sparse backsubstitution $Rx = b$, given $n$, the dimensions of dense $x$ and $b$ as well as of the sparse $R$ and $nnz$, the number of nonzero entries in $R$?
2
votes
2answers
75 views

Simplfy a complex matrix into a real one

I encounter systems of linear complex equations (At most 3 equations) in my circuit analysis course. The calculator I am using is Casio fx-991ES and it only accepts real elements when in matrix or ...
0
votes
2answers
98 views

System of linear equations problem

Consider the system of linear equations: \begin{align} \begin{cases} x+ay=1\\[2ex] bx+5y=2, \end{cases} \end{align} where $a$ and $b$ are parameters. (a) Determine the conditions on $a$ and $b$ to ...
1
vote
4answers
92 views

Reducing the System of linear equations

\begin{align*} x+2y-3z&=4 \\ 3x-y+5z&=2 \\ 4x+y+(k^2-14)z&=k+2 \end{align*} I started doing the matrix of the system: $$ \begin{pmatrix} 1 & 2 & -3 & 4 \\ 3 & -1 & 5 ...
0
votes
0answers
25 views

Is there a non-zero solution to this system of equations?

I have a system of equations. $\mu_1p_1=(1-p_1)\sum_{i=1, i\ne 1}^N \lambda_{i1}p_i$ $\mu_2p_2=(1-p_2)\sum_{i=1, i\ne 2}^N \lambda_{i2}p_i$ and so on. The values $\mu_k$ and $\lambda_{kl}$ are ...
1
vote
0answers
84 views

Solving a Large band system using Gauss-Seidel Iteration

Sorry for my english. I have to solve the following band system using Gauss-seidel iteration program in matlab. $$ \begin{array}{cccccc} 12x_1&-2x_2&+x_3&&&=&5\\ ...
1
vote
1answer
52 views

How to convert this equation into a matrix form

$$F(x)=aF(x-k+1)+bF(x-k+2)+cF(x-k+4)$$ where $F(x)=1$ if $x<k$. $a,b,c,k$ are known (and positive) and $x$ is chosen. I want to solve this recurrence using a matrix but don't really know how to ...
1
vote
0answers
42 views

Finding Fundamental Matrix of Differential Equations with Periodic Coefficients

So I am supposed to find the fundamental matrix of $x'_1=(1+2\cos(2t))x_1+(1-2\sin(2t))x_2$ $x'_2=-(1+2\sin(2t)x_1+(1-2\cos(2t))x_2$ The book suggests using the transformation $x=Q(t)\hat{x}$ ...
0
votes
0answers
49 views

Iteration to Solve Unit Row Diagonally Dominant System

Given a matrix is unit row diagonally dominant $a_{ii}=1>\sum^n_{j=1,j\neq i} |a_{ij}|, \hspace{4mm} 1 \leq i \leq n$, prove that the following iteration will solve $Ax=b$ in the limit. $for ...
1
vote
2answers
78 views

Why can't you swap rows in the matrix for a system of linear differential equations?

If you are given a Matrix A, and then asked to solve the initial value problem x'=Ax, why can one not swap rows before starting the problem. I tried it with a 3x3 matrix on wolfram alpha and got two ...
0
votes
0answers
154 views

Kronecker-Capelli Theorem for system of congruences

Let $p$ be some prime. Given a system of linear congruences, \begin{align} m_1 x + n_1 y &\equiv c_1 \quad (mod\, p)\\ m_2 x + n_2 y &\equiv c_2 \quad (mod\,p)\\ \ldots \\ m_d x + n_d y ...
0
votes
0answers
15 views

Linear independent vectors and Base of linear Hull

Assignment Decide whether the following vectors are linear independent and give the Base of its linear Hull of the family $(v_i)_{i \in I}$: $$v_1 = \begin{pmatrix} 4 \\ 9 \\ 5 \end{pmatrix}, v_2 = ...
2
votes
1answer
201 views

Gauss-Jordan Elimination to solve for variables

I have the following linear system: $$x + 2y - 3z = 4$$ $$3x - y + 5z = 2$$ $$4x + y + (s^2 - 14)z = s+2$$ Im trying to solve for $s$ to figure out how many solutions it has (if any). I know how to ...
-2
votes
1answer
57 views

Determine $b$ where the system has a solution

Determine the value of $b$ for which the system $$\begin{align} x_1 + 4x_2 − 3x_3 + 2x_4 &= 2\\ 2x_1 + 7x_2 − 4x_3 + 4x_4 &= 3\\ −x_1 − 5x_2 + 5x_3 − 2x_4 &= b\\ 3x_1 +10x_2 − 5x_3 + ...
0
votes
1answer
1k views

find all the values of a and b so that the system has a) no solution b) 1 solution c) exactly 3 solutions and 4) infinitely many solutions

$$\begin{cases} &x &- &y &+ &2z &= 4 \\ &3x &- &2y &+ &9z &= 14 \\ &2x &- &4y &+ &az &= b \end{cases} $$ I know that $a$ and ...
1
vote
1answer
130 views

Rewriting the simplified google algorithm in linear algebra form

I have the expression for the rank ($x_{i}$) of a page $i$ in an internet with $n$ sites, each site contains $n_{i}$ links to other sites and is linked to by the pages $L_{i}\subset\{1,\dots,n\}$. The ...
1
vote
1answer
201 views

Solving systems of linear equations with an unknown 'a' using matrices and elementary row operations

Came across this one the other day... while I can narrow 'a' down I can't seem to find an exact/ optimised figure. For example 'a' cannot equal 1/3, 'a' must be less than 0.5... Anyway, here's the ...