0
votes
1answer
38 views

Invertible Matrices Proof

Given that B is an invertible matrix and $B^3 + B^4 + B^7 = I$, find an expression for $B^{-1}$ in terms of only $B$. (where $I$ is an identity matrix) $B$ is a matrix that is $n \times n$.
0
votes
0answers
33 views

show by using leibniz formula

There are given $ r, s,n \in\mathbb N$ and $r+s=n$. It also given $A \in M_{r,K} $, $B \in M_{r\times s,K} $ and $C \in M_{s,K} $. Let $M$ be the matrix $\begin{bmatrix}A & B\\0 & ...
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
2answers
21 views

Identify Orthogonal Proj. and Reflection within given choice of Matrices.

The problem states that out of five given matrices, one represents an Orthogonal Projection onto a line and another a Reflection about a line; I'm supposed to identify them. Rather than list the ...
4
votes
2answers
321 views

Eigenvalues and Eigenspace Question

Thank you ahead of time for the help, I am having a problem with part $4$. I understand parts $1$ and $2$ and $3$ and have solved them but I cant seem to understand $4$. If someone could help me out, ...
1
vote
1answer
36 views

Are these two matrices equivalent?

I am supposed to row reduce a matrix to reduced row echelon form. $$ \begin{bmatrix} 1 & 2 & 4 & 8\\ 0 & 0 & 1 & 4\\ 0 & 0 & 0 & 0 \end{bmatrix} $$ I have tried ...
6
votes
1answer
98 views

Existence of $p \times p $ matrices $A$ and $B$ over the field $\mathbb F_p$, $p$ prime, such that $AB-BA=I$. [duplicate]

Let $p$ be a prime number. Prove or disprove that there exists $p\times p$ matrices $A$ and $B$ over a field $\mathbb F_p$ with $AB-BA = I$. With the aid of MAPLE i was able to find out that ...
0
votes
1answer
12 views

Question about Joint spectral radius.

Given a bounded set $\mathcal A\subset \Bbb R^{n x n}$. The joint spectral radius is given by: $\sigma(\mathcal A)$=$limsup_{m\to\infty}(sup_{A\in\mathcal A^m} \rho(A))$ where $\rho$ is the normal ...
0
votes
1answer
35 views

Is there a subspace of $M^R_{2x3}$ that is isomorphic to $R_4[x]$?

Is there a subspace of $M^R_{2x3}$ that is isomorphic to $R_4[x]$? For example, Can I say that $M^R_{2x2}$ is a subspace of $M^R_{2x3}$ so it can be isomorphic to $R_4[x]$ ? (because they have ...
0
votes
1answer
28 views

Gauss-Jordan Method

I keep getting the wrong set of solutions can someone help me. I know that when using the Gauss-Jordan method, the rules that I must follow can be applied in a variety of different procedures then why ...
1
vote
1answer
44 views

Matrix with unknown coefficients, finding another basis

let $(e_1,e_2,...,e_5)$ canonical basis of $R^5$, $V=(a,b,c,d,e)\in R^5$ with $V\neq(0,0,0,0,0)$. we consider $f:R^5\to R^5$ and its matrix : $$Mat(f) = M= \begin{pmatrix} ...
2
votes
2answers
40 views

M is real anti-symmetric matrix, prove that exp(M) is isometry

M is nxn real anti-symmetric matrix.I need to prove that exp(M) is isometry. Could anyone give me any hint , I don't have any approach to this question. thank you
1
vote
1answer
32 views

What's the connection between rank of matrix and $0$ eigenvalue?

My matrix B is nxn and know nothing about if diagonalizble, but I know that rank B = 1. Therefore the geometric multiplicity of λ=0 as an eigenvalue is n-1. But by knowing the rank is 1, can I say ...
3
votes
3answers
46 views

$A$ matrix, $+i, -i$ are eigenvalues.

If matrix $A$ is a square matrix. And $A$'s characteristic polynomial is $p(t) = t^2 + 1$. It's not necessary true that $A$ is non-singular right? Because, the eigenvalues are $i,-i$. and if $\dim(A)= ...
1
vote
0answers
25 views

Ax = B, group of columns of A.

Is there a matrix $A (n \times n)$ Over field $F$ and $b \in F^n$ has non-trivial solution to the equation $Ax=B $ ? Well, In the answer it is written that because that the set of the columns of ...
0
votes
0answers
12 views

Interpreting & Analysing a Transitional Matrix

How do you interpret such a problem Are we expect to add the rows, and that would be the one with larger number of goats in the long term. Therefore A(row 1) and b(row 2)... therefore the answer is ...
2
votes
1answer
22 views

If the first r columns of U are linearly independent, then so are the first r columns of A?

Let $U$ be a row echelon form of a square matrix $A$. If the first $r$ columns of $U$ are linearly independent, then should the first $r$ columns of $A$ be linearly independent? In my opinion, "Yes" ...
3
votes
1answer
22 views

Matrix $A$ with characteristic polynomial

Given: Matrix $A$ with characteristic polynomial $p(x) = (x+3)^2(x-1)(x-5)$ Also given: $\rho(A+2I) + \rho(A+3I) + \rho(A-5I) = 9$ (btw $\rho$ means rank of the matrix) Prove: $A$ is ...
1
vote
1answer
45 views

How can I find $T^{-1}(x,y,z)$ (inverted matrix) of a linear operator $T:V_3 \to V_3$

How can I find $T^{-1}(x,y,z)$ (inverted matrix) of a linear operator $T:V_3 \to V_3$, which matrix relative to the basis: $A=\{ (1,0,0), (1,1,0), (1,1,1)\}$ is: $$T_A= \begin{bmatrix} 2 &0 ...
3
votes
6answers
374 views

How can I prove that a square matrix is invertible if it satisfies this polynomial equation?

For a 3x3 matrix $C$, it is given that $$C^3+I=3C^2-C$$ I am then required to prove that $C$ is invertible. I have attempted a proof, below, but I am not sure it is valid or if there is a better ...
1
vote
1answer
20 views

The similarity of the block matrices

Let $\mathbb{F}$ be a field,and let $A,B,C$ be matrices over $\mathbb{F}$ of respective sizes $n\times n , k\times k, $and $n\times k$. put $M=\begin{bmatrix} A&0 \\ 0&B ...
-1
votes
2answers
49 views

A question on Rank and trace of a special matrix [closed]

I want to share the following question which was asked in a competitive exam: For a fixed positive integer $n\geq 3$, let $A$ be the $n\times n$ matrix defined by $A=I-\dfrac{1}{n}J$, where $J$ is ...
2
votes
1answer
74 views

Some questions on Nilpotent matrix [closed]

Q & A style. Just wanted to share the following question which came in a competitive exam and so college level maths students may find it useful: A non-zero matrix $A\in M_n(\mathbb{R})$ is said ...
0
votes
1answer
37 views

How to compute the eigenvalue condition number of a matrix

How to compute the eigenvalue condition number, $\kappa(4,A)$, of a matrix $A$ $$A = \begin{bmatrix} 4 & 0 \\ 1000 & 2\end{bmatrix}$$ I am a bit stuck on how to proceed solving this problem ...
0
votes
0answers
19 views

Transformation matrices and hermitian/unitary/normal/… matrices

I need some help with the following - have I done the correct things or how can I solve the task? Let $f \in End(V)$, V a unitary space $\mathbb{C}^3$ given by: $A_{\alpha \beta} (f) = \frac{1}{7} ...
0
votes
1answer
47 views

Diagonalizable A, computing fast.

I have $A =$ $ \begin{pmatrix} a & 0 & 0 \\ b & 0 & 0 \\ 1 & 2 & 1 \\ \end{pmatrix} $ I know that this matrix $A$ is diagonalizable when ...
1
vote
2answers
16 views

Scalar product, undefined matrix multiplications and order of operations

I am curious about the order of operations in linear algebra. Let $\mathbf{u},\mathbf{v} \in \mathbb{R}^{n}$ and $\mathbf{A},\mathbf{B}\in \mathbb{R}^{m \times m}$, with $m\neq n$ and $\mathbf{u} ...
0
votes
1answer
56 views

Tips on how I would find the transition matrix for the following phenomenon?

how would I go about finding the transition matrix for the following phenomenon (which can be modeled as a Markov process)? Any hints or advice is appreciated! During a study break, a student's ...
1
vote
4answers
367 views

Linear algebra, power of matrices

$P^{-1}AP = \begin{pmatrix} -1 & 1 & 0 & 0 \\ 0 & -1 & 1 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 2 \\ \end{pmatrix} $ with $P= ...
1
vote
2answers
33 views

Determine the values of $k$ so that the following linear system has unique, infinite and no solutions.

Determine the values of $k$ so that the following linear system has a unique solution, infinite solutions and no solution. $2x + (k + 1)y + 2z = 3$ $2x + 3y + kz = 3$ $3x + 3y − 3z = 3$ I have ...
0
votes
0answers
38 views

Division by zero while generating the matrix

I got the formula and I need to create the lineral-equation system matrix using it and then solve it. I decided to use wolfram mathematica to do this. I wrote the code. And when N and M are $> ...
3
votes
1answer
88 views

Eigenvalues of a symmetric matrix with Lagrange multipliers

Problem: Using Lagrange multipliers, prove that all symmetric matrices $A \in \mathbb{R}^{n \times n}$ have all real eigenvalues. Proof: Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}$ defined by ...
1
vote
1answer
79 views

For which $x,y,z,w$ is matrix $A$ orthogonal/unitary?

given is $A = \frac{1}{2} \begin{pmatrix} x & 1 & 1 & 1 \\ y & 1 & -1 & 1 \\ z & 1 & -1 & -1 \\w & 1 & 1 & -1 \end{pmatrix} $ How do I have to chose ...
0
votes
1answer
40 views

Solve Matrix: How many trips can be made?

We have the matrix: $$\mathbf{M}=\begin{bmatrix}0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0\end{bmatrix}$$ The matrix ...
12
votes
5answers
272 views

Prove that $A^k = 0 $ iff $A^2 = 0$

Let $A$ be a $ 2 \times 2 $ matrix and a positive integer $k \geq 2$. Prove that $A^k = 0 $ iff $A^2 = 0$. I can make it to do this exercise if I have $ \det (A^k) = (\det A)^k $. But this ...
1
vote
2answers
45 views

Prove that if $B=P^{-1}AP$, then $q(B)=P^{-1}q(A)P$

Is it possible to prove this using a similarity invariant? For example showing that $$\det(q(B))=\det(q(A))$$
0
votes
1answer
42 views

Finding the representing matrix with respect to the standard basis.

Let $B=[(1,0,0),(1,2,0),(1,2,3)]$ be the basis for $\mathbb{R}^3$. Let $T:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be a linear transformation, such that its representing matrix with respect to basis $B$ ...
0
votes
1answer
29 views

what does this question about a matrix mean?

here is a question says : what does that mean ? I did my best to solve this question myself but i didn't find a way to solve it is this question possible or there is something else that i don't ...
1
vote
1answer
45 views

Find the last column of a matrix. Find the matrix.

$A\left[\begin{matrix} 1 & 0 \\ 0 & 0 \\ 1 & 1 \end{matrix} \right]$ = $\left[\begin{matrix} 2 & 3 \\ -1 & 0 \\ 5 & -7 \\ 0 & 6 \end{matrix} \right]$ (1)Find the last ...
4
votes
0answers
76 views

Diagonalization

I'm having difficulty understand some questions. I will highlight the terms I do not understand. Question 1: Let $A =\begin{pmatrix} 1& -2 \\ 1& 3 \end{pmatrix}$ For the matrix $A$, ...
0
votes
0answers
21 views

A question on companion matrix

Let $A^*$ be the companion matrix of matrix $A$, where $A$ is $n\times n$ type. Then $$|-A^*|=(-1)^{n}|A^*|.$$ Is it right? I'm not sure. Thanks for your help! Thanks ahead:)
1
vote
0answers
28 views

About Jordan-Chevalley decomposition

I have this problem: Let $K$ be a field. Let $J\in M_n(K)$ a Jordan matrix. Prove that there exists a diagonal matrix $D$ and a nilpotent matrix $N$ such that $J=D+N$ and $DN=ND$. I saw that this ...
0
votes
1answer
41 views

What does the question in the attached image mean?

here is a question : http://upimage.us/server/php/files/math%20Q%20%281%29.jpg what does " echelon form " mean ?
1
vote
2answers
45 views

If $AX=XA$ for all $X$, then $A = \alpha I$ for some $\alpha$

Let $A$ be a $2 \times 2$ real matrix such that $AX=XA$ for all $2 \times 2$ real matrices $X$. Show that $A= \alpha I$ for some $\alpha ∈R.$ I am absolutely stuck, i thought $X$ and $A$ are ...
2
votes
2answers
35 views

Minimizing Frobenius norm for two variables

I need to minimize squared Frobenius norm: $\|\mathbf{A} - \mathbf{x}\mathbf{y}^T\|_F^2$. Namely I need to prove that for this norm to reach minimum $\mathbf{x}$ should be eigenvector of ...
0
votes
1answer
34 views

Gram-Schmidt method and matrices help please!

How would I use the method of Gram-Schmidt to obtain an orthonormal basis for the column space of the matrix? Any help is appreciated!
2
votes
2answers
41 views

Gershgorin discs and norm of a matrix

Find a matrix, where the estimation of eigenvalues with the help of Gershgorin discs is a, the same as b, worse as the estimation with the help of the norm of the matrix ($||A||_\infty$) So, yes, ...
1
vote
0answers
36 views

problems on define set with polynomials

I'm trying to say set A is the set of nonnegative integers that not of this two forms $3x^2 + (6y-4)x - y\ $ and $\ 3x^2 + (6y-2)x + (y - 1)$, for example: $4=3 \cdot1^2+(6 \cdot1-4) \cdot 1-1\ $ is ...
1
vote
1answer
24 views

Prove $||A||_2 = max_{x \neq 0, y \neq 0}\{\frac{|\langle y,Ax \rangle|}{||x||\cdot ||y||}\}$ for $A\in \mathbb{K}^{n\times m}$

I'd like to prove that the spectral norm of a matrix that is not necessarily square can be written as the following subordinate norm $||A||_2 = max\{\frac{|\langle y,Ax \rangle|}{||x||\cdot ||y||}, y ...
2
votes
1answer
42 views

Positive semi-definite Matrix and its eigenvalues (Please help checking/ improving my presentation)

Let $A$ and $B$ are two $n \times n$ Hermitian matrices . Suppose $A-B$ is positive semidefinite. (a) Show that $\lambda_k(A) \geq \lambda_k(B)$ for $k=1,2,\dots ,n,$ where $\lambda_i(A)$ and ...