# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $$\det(q(B))=\det(q(A))$$
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### 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 $B$ ...
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### 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 ...
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### 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 ...
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### 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 $A$, ...
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### 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:)
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 ...