0
votes
3answers
29 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
35 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
23 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
28 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 ...
0
votes
0answers
23 views

Matrices and solving 3 variable systems

The problem is as follows: You sell tickets at school for fundraisers. You sold car wash tickets, silly string fight tickets and dance tickets – for a total of 380 tickets sold. The car wash ...
5
votes
1answer
94 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
46 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
23 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
43 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
62 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
32 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
71 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
37 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
54 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
14 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
44 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
52 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
31 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
28 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
42 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
193 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
20 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
72 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
90 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
82 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
40 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
45 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
71 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
138 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
14 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
198 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
56 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
961 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
190 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 ...
2
votes
1answer
173 views

System of Linear Equations (3x6 matrix, parametric answer)

Solve the system \begin{array}{r@{}r@{}r@{}r@{}r@{}r@{}r@{}r} x_1 & - 2 x_2 & - 2 x_3 & & + 5 x_5 & - 4 x_6 & = & -1 \cr & & & - ...
1
vote
2answers
49 views

Finding the amount of solutions in a 3 equation solution

So, I'm not really sure how to calculate the amount of solutions for a system with 3 equations. All I know is that it has something to do with matrices and the discriminant, but I'm not sure where to ...
2
votes
2answers
87 views

Solving a system of three linear equations with three unknowns

Is my working correct or am I completely wrong? Have I missed anything out? Any feedback is appreciated. Question: Consider the following system of equations $2x + 2y + z = 2$ $−x + 2y − z = −5$ ...
0
votes
1answer
656 views

Consistency of a System of linear equations

Test the consistency of the system of linear equations $$\begin{align} 4x-5y+z & =2 \\ 3x+y-2z& = 9 \\ x+4y+z& =5\end{align}$$
0
votes
1answer
400 views

Simultaneous Equations That Should Be Inconsistent Has a Unique Solution

Find the values of $k$ for which the simultaneous equations do not have a unique solution for $x, y$ and $z$. Also show that when $k = -2$ the equations are inconsistent $$kx + 2y +z =0$$ $$3x + 0y ...
0
votes
2answers
166 views

Solving Linear Systems of Differential Equations - Phase Portraits

I'm working on trying to find the phase portrait for a differential equations such that $$ A = \begin{matrix} a_1 & a_2 \\ 0 & a_3 \end{matrix}$$ so that $$x_1'(t) = a_1x_1+a_2x_2 \text { and ...