0
votes
0answers
16 views

System solving with Substitution and Matrices

My class was able to produce solutions using Substitution on the following System: $$ \left\{ \begin{array}{c} x+y+z=0 \\ 2x+3y+2z=-1\\ x-y+z=2 \end{array} \right. $$ The solution was: x = 1, y = ...
0
votes
1answer
35 views

Find the standard matrix of the linear transformation

Suppose there is a linear transformation $T:\mathbb{R^2} \rightarrow \mathbb{R^2}$ such that $$T\left( \begin{array}{ccc}2 \\ 1 \end{array} \right)=\left( \begin{array}{ccc}1\\ 4 \end{array} ...
1
vote
1answer
53 views

Rational solution to AX=0

Let $\mathcal{M}_{n,p}(\mathbb{K})$ be the set of matrices $n\times p$ with coefficients in $\mathbb{K}$. Let $A\in\mathcal{M}_{n,p}(\mathbb{Q})$. We suppose there exists a non zero solution ...
2
votes
0answers
50 views

Computing a “cheap” upper bound on the norm of the solution to a linear system

Consider the linear system $A x = b$, where $A$ is an invertible, $n \times n$, real matrix. I would like to compute a "cheap" upper bound on the (p-)norm of the solution; i.e. $\|x\|_p$. One can, of ...
0
votes
1answer
32 views

Which complex vector multiplied by its conjugate returns the identity matrix

I am trying to find (in case there is any) which complex vector $n$ of 2 dimensions, multiplied by its conjugate transpose, returns a diagonal matrix. $n = [a, b]^T = [a_1+ja_2, b_1+jb_2]^T$ ...
1
vote
1answer
32 views

If we know nullspace of matrix, how to find reduced row echelon form of that matrix?

vectors u = [4 1 0 0] and v = [1 0 2 1] form a base of nullspace of matrix $$ A\in M_{5,4}(R) $$ Find a reduced row echelon form of Matrix A. Since $ n-r = dimN(A) $ we know we got two base ...
0
votes
1answer
45 views

Solve the linear system by Gauss - Jordan elimination

$$ \begin{align} x& - y + 2z - w &= -1\\ 2x& + y - 2z - 2w &= -2\\ -x& + 2y - 4z + w &= 1\\ 3x& -3w &= -3 \end{align} $$ ...
0
votes
1answer
43 views

Solve the linear system by Gaussian elimination

$\begin{cases}-2b+3c=1 \\ 3a+6b-3c=-2 \\ 6a+6b+3c=5\end{cases}$ I got an inconsistent linear system with the third row being 0 0 0 6. May someone please verify if I am right? I looked it over.
5
votes
4answers
404 views

Linear Algebra - four “true or false” questions about matrices and linear systems

I'm reviewing for my linear algebra course, and have four "true or false" questions that I'm struggling to prove. I've included my approach to the solutions in brackets below them: 1) If $A^2 = B^2$, ...
0
votes
0answers
24 views

Matrix, Gauss-Jordan Method

I have a application problem for math and I am unable to get all my system of equations. I have two of three. Celia had one hour to spend at the athletic club, where she will jog, play handball, and ...
0
votes
1answer
32 views

Solve $3$ variables using $4$ equations where $1$ equation contains $3$ variables

Suppose we are given the system of equations $$\alpha_1A+\beta_1B+\gamma_1C=x$$ $$\alpha_2A+\beta_2B+\gamma_2C+\theta_2D=y$$ $$\alpha_3A+\beta_3B+\gamma_3C+\theta_3D=z$$ where ...
0
votes
1answer
43 views

How to find system of equations from solution space

I have to find homogeneous system of linear equations whose solution space is: V = span((1,-2,4,3),(1,-1,6,4),(3,-8,8,3)). First I found vectors were linearly dependent, so I discarded the third ...
-2
votes
2answers
32 views

System of linear equation matrix? [duplicate]

How would I do this question. Determine the value(s) of $h$ such that the matrix is augmented of a consistent linear system. My matrix \begin{bmatrix} 1&h&4\\ 3&6&8 \end{bmatrix} I ...
1
vote
0answers
67 views

Find the fundamental matrix of a system of ODEs?

To linearize a system, in one of the steps I am required to find the fundamental matrix $\Phi$(t) of a system such that $\Phi$(0)=I. The example system my professor used: $\dot{x} = x - y - x^3 - ...
0
votes
0answers
37 views

Solving system of equations in rationals

Do there exist solutions to solve system of $n-2$ equations with $n-2$ variables where $n$ is fixed even integer and $a_i,b,c\in\mathbb{Q},i\in\{0,1,2,\cdots,n-5\}$ $$\left\{ ...
0
votes
1answer
28 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 ...
1
vote
0answers
21 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
36 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
241 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
23 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
34 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
42 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
60 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
41 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
34 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
110 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
51 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
73 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
25 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
52 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
67 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
63 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
21 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
77 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
76 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
63 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
52 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
18 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
82 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
62 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
84 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
40 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
30 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
48 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
542 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
31 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
101 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
104 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
94 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 ...