1
vote
1answer
57 views

If $AB+BA=0$ and $B=AX+XB$, then $B$ is nilpotent.

Suppose $A,B,X \in M_n(\mathbb{R})$ and that $AB+BA=0$ and $B=AX+XA$. Prove that $B$ is a nilpotent matrix.
3
votes
2answers
95 views

Interesting determinant: Let $A$ be an $n$ by $n$ matrix with entries $a_{i,j}$ given that $a_{i,j}=2$ if $i=j$

Let $A$ be an $n$ by $n$ matrix with entries $a_{i,j}$ given that $a_{i,j}=2$ if $i=j$, $a_{i,j}=1$ if $i-j\equiv\pm2\pmod n$, and $a_{i,j}=0$ otherwise. Find $\det A$. It seems that the ...
0
votes
2answers
47 views

Given $a > b+c$, $e>d+f$, and $i>g+h$, can the quantity $a(ei-hf) + b(-di+fg) - c(dh+eg)$ ever be zero?

Given positive reals $a > b+c$, $e>d+f$, and $i>g+h$, can the quantity $a(ei-hf) + b(-di+fg) - c(dh+eg)$ ever be zero?
4
votes
2answers
543 views

how to prove following matrix is invertible? [duplicate]

how to prove A is invertible or $\ detA\neq 0$ $$A=\begin{pmatrix} \frac11 & \frac12 & \frac13 & \cdots & \frac1n \\ \frac12 & \frac13 & \frac14 & \cdots & ...
2
votes
1answer
61 views

If $H$ has an $a\times b$ submatrix of all $1$s, please prove that $ab\le n$.

Let $H$ be an $n\times n$ matrix with entries $\pm1$. Its rows are mutually orthogonal. If $H$ has an $a\times b$ submatrix of all $1$s, please prove that $ab\le n$.
4
votes
2answers
205 views

Prove that function is bijective

Let $n \in \mathbb{N} \setminus \{ 0 \} $ and $A \in M_n(\mathbb{R})$ with $m \in \mathbb{N} \setminus \{ 0 \}$ as $A^m= \alpha \times I_n$, with $ \alpha \in \mathbb{R} \setminus \{ -1,1 \}$. ...
5
votes
3answers
213 views

Show that $\operatorname{rank}(A^2+A+I_3)=1$

If $A \in M_3(\mathbb{R}), A \ne I_3 $ and $A^3=I_3$ Show that $\operatorname{rank}(A^2+A+I_3)=1$. What I have reached so far is that $\operatorname{rank}(A-I_3)+\operatorname{rank}(A^2+A+I_3)\le ...
4
votes
2answers
71 views

let $A\in M_n\mathbb R$ .how prove these statements with following condition?

Assume $A\in M_n(\mathbb{R})$, $A\neq 0$ such that: \begin{align*} A=(a_{ij}),\ 1\le i,j\le n,\\ a_{ik}a_{jk}=a_{kk}a_{ij},\ \forall\, i,j \end{align*} How to prove that: ...
2
votes
1answer
79 views

Assume $A_1,A_2,…,A_n\in M_{m×m}(F)$ that satisfy the following conditions, how to prove that $A_1A_2…A_n=0$?

Assume $A_1,A_2,...,A_n\in M_{m×m}(F)$ (where $F$ is a field) such that $A_jA_i=A_iA_j$ $A_i^2=0, \;\;\forall 1\leq i \leq n.$ If $m\lt2^n$ then how to prove that $A_1A_2...A_n=0.$ Thanks in ...
10
votes
2answers
230 views

Let the matrix $A=[a_{ij}]_{n×n}$ be defined by $a_{ij}=\gcd(i,j )$. How prove that $A$ is invertible, and compute $\det(A)$?

Let $A=[a_{ij}]_{n×n}$ be the matrix defined by letting $a_{ij}$ be the rational number such that $$a_{ij}=\gcd(i,j ).$$ How prove that $A$ is invertible, and compute $\det(A)$? thanks in advance
26
votes
3answers
557 views

Square matrices satisfying certain relations must have dimension divisible by $3$

I saw this tucked away in a MathOverflow comment and am asking this question to preserve (and advertise?) it. It's a nice problem! Problem: Suppose $A$ and $B$ are real $n\times n$ matrices with ...
2
votes
2answers
164 views

let A,B be complex matrics and $2A(B-A)=A+B$ how prove $AB=BA$

let $A,B\in M_n(\mathbb C)$ $\mathbb C$ is complex field such that $$2A(B-A)=A+B$$ how prove $AB=BA$ thanks in advance
5
votes
3answers
381 views

prove that $\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$

If $A$ is a $m \times n$ matrix and $B$ a $n \times k$ matrix, prove that $$\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$$ Also show when equality occurs.
3
votes
2answers
96 views

Putnam type question: Invertible matrix

Are the following matrices invertible? (1) $A= (a_{ij})_{2003 \times 2003}$, where $a_{ii}=2003, a_{ij}=1$ for $i \not=j$. (2) $B= (b_{ij})_{n \times n }$ with $b_{ii}= \pi$ and $b_{ij} \in ...
2
votes
1answer
248 views

Applying a Function to Square Matrices

Moderator Note: This question is from a contest which ended 1 Dec 2012. Consider a polynomial $f$ with complex coefficients. Call such $f$ broken if we can find a square matrix $M$ such that $M ...
1
vote
2answers
252 views

Ordered quadruples in a grid

Moderator Note: This question is from a contest which ended on 22 Oct 2012. Consider $(\alpha_1, \alpha_2, \alpha_3, \alpha_4)$ such that the ordered quadruple satisfies the following: ...
11
votes
1answer
217 views

Proving a certain determinant $\left|\det A\right|$ is complete square

Consider the following matrix $$ A_{ij}= \begin{cases} 1\quad\text{ if }\space (i+j)\space\text{ is prime,}\\ 0\quad\text{ otherwise.} \end{cases} $$ How can one prove that $\left|\det A\right|$ is a ...