Found problems: 93
2009 Miklós Schweitzer, 11
Denote by $ H_n$ the linear space of $ n\times n$ self-adjoint complex matrices, and by $ P_n$ the cone of positive-semidefinite matrices in this space. Let us consider the usual inner product on $ H_n$
\[ \langle A,B\rangle \equal{} {\rm tr} AB\qquad (A,B\in H_n)\]
and its derived metric. Show that every $ \phi: P_n\to P_n$ isometry (that is a not necessarily surjective, distance preserving map with respect to the above metric) can be expressed as
\[ \phi(A) \equal{} UAU^* \plus{} X\qquad (A\in H_n)\]
or
\[ \phi(A) \equal{} UA^TU^* \plus{} X\qquad (A\in H_n)\]
where $ U$ is an $ n\times n$ unitary matrix, $ X$ is a positive-semidefinite matrix, and $ ^T$ and $ ^*$ denote taking the transpose and the adjoint, respectively.
2005 Romania National Olympiad, 1
Let $n\geq 2$ a fixed integer. We shall call a $n\times n$ matrix $A$ with rational elements a [i]radical[/i] matrix if there exist an infinity of positive integers $k$, such that the equation $X^k=A$ has solutions in the set of $n\times n$ matrices with rational elements.
a) Prove that if $A$ is a radical matrix then $\det A \in \{-1,0,1\}$ and there exists an infinity of radical matrices with determinant 1;
b) Prove that there exist an infinity of matrices that are not radical and have determinant 0, and also an infinity of matrices that are not radical and have determinant 1.
[i]After an idea of Harazi[/i]
2012 Pre-Preparation Course Examination, 3
Suppose that $T,U:V\longrightarrow V$ are two linear transformations on the vector space $V$ such that $T+U$ is an invertible transformation. Prove that
$TU=UT=0 \Leftrightarrow \operatorname{rank} T+\operatorname{rank} U=n$.
2006 Romania National Olympiad, 2
We define a [i]pseudo-inverse[/i] $B\in \mathcal M_n(\mathbb C)$ of a matrix $A\in\mathcal M_n(\mathbb C)$ a matrix which fulfills the relations
\[ A = ABA \quad \text{ and } \quad B=BAB. \]
a) Prove that any square matrix has at least a pseudo-inverse.
b) For which matrix $A$ is the pseudo-inverse unique?
[i]Marius Cavachi[/i]
2011 Romania National Olympiad, 4
[color=darkred]Let $A\, ,\, B\in\mathcal{M}_2(\mathbb{C})$ so that : $A^2+B^2=2AB$ .
[b]a)[/b] Prove that : $AB=BA$ .
[b]b)[/b] Prove that : $\text{tr}\, (A)=\text{tr}\, (B)$ .[/color]
2008 District Olympiad, 1
If $A\in \mathcal{M}_2(\mathbb{R})$, prove that:
\[\det(A^2+A+I_2)\ge \frac{3}{4}(1-\det A)^2\]
2010 VJIMC, Problem 2
If $A,B\in M_2(C)$ such that $AB-BA=B^2$ then prove that
\[AB=BA\]
2013 Romania National Olympiad, 1
Given A, non-inverted matrices of order n with real elements, $n\ge 2$ and given ${{A}^{*}}$adjoin matrix A. Prove that $tr({{A}^{*}})\ne -1$ if and only if the matrix ${{I}_{n}}+{{A}^{*}}$ is invertible.
1984 IMO Longlists, 64
For a matrix $(p_{ij})$ of the format $m\times n$ with real entries, set
\[a_i =\displaystyle\sum_{j=1}^n p_{ij}\text{ for }i = 1,\cdots,m\text{ and }b_j =\displaystyle\sum_{i=1}^m p_{ij}\text{ for }j = 1, . . . , n\longrightarrow(1)\]
By integering a real number, we mean replacing the number with the integer closest to it. Prove that integering the numbers $a_i, b_j, p_{ij}$ can be done in such a way that $(1)$ still holds.
1986 IMO Longlists, 46
We wish to construct a matrix with $19$ rows and $86$ columns, with entries $x_{ij} \in \{0, 1, 2\} \ (1 \leq i \leq 19, 1 \leq j \leq 86)$, such that:
[i](i)[/i] in each column there are exactly $k$ terms equal to $0$;
[i](ii)[/i] for any distinct $j, k \in \{1, . . . , 86\}$ there is $i \in \{1, . . . , 19\}$ with $x_{ij} + x_{ik} = 3.$
For what values of $k$ is this possible?
2011 District Olympiad, 2
Consider the matrices $A\in \mathcal{M}_{m,n}(\mathbb{C})$ and $B\in \mathcal{M}_{n,m}(\mathbb{C})$ with $n\le m$. It is given that $\text{rank}(AB)=n$ and $(AB)^2=AB$.
a)Prove that $(BA)^3=(BA)^2$.
b)Find $BA$.
1999 IMC, 5
Suppose that $2n$ points of an $n\times n$ grid are marked. Show that for some $k > 1$ one can select $2k$ distinct marked points, say $a_1,...,a_{2k}$, such that $a_{2i-1}$ and $a_{2i}$ are in the same row, $a_{2i}$ and $a_{2i+1}$ are in the same column, $\forall i$, indices taken mod 2n.
2011 District Olympiad, 3
Let $A,B\in \mathcal{M}_2(\mathbb{C})$ two non-zero matrices such that $AB+BA=O_2$ and $\det(A+B)=0$. Prove $A$ and $B$ have null traces.
2006 Romania National Olympiad, 1
Let $A$ be a $n\times n$ matrix with complex elements and let $A^\star$ be the classical adjoint of $A$. Prove that if there exists a positive integer $m$ such that $(A^\star)^m = 0_n$ then $(A^\star)^2 = 0_n$.
[i]Marian Ionescu, Pitesti[/i]
2010 IMC, 4
Let $A$ be a symmetric $m\times m$ matrix over the two-element field all of whose diagonal entries are zero. Prove that for every positive integer $n$ each column of the matrix $A^n$ has a zero entry.
2007 Romania National Olympiad, 3
Let $n\geq 2$ be an integer and denote by $H_{n}$ the set of column vectors $^{T}(x_{1},\ x_{2},\ \ldots, x_{n})\in\mathbb{R}^{n}$, such that $\sum |x_{i}|=1$.
Prove that there exist only a finite number of matrices $A\in\mathcal{M}_{n}(\mathbb{R})$ such that the linear map $f: \mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ given by $f(x)=Ax$ has the property $f(H_{n})=H_{n}$.
[hide="Comment"]In the contest, the problem was given with a) and b):
a) Prove the above for $n=2$;
b) Prove the above for $n\geq 3$ as well.[/hide]
2006 Iran MO (3rd Round), 3
Suppose $(u,v)$ is an inner product on $\mathbb R^{n}$ and $f: \mathbb R^{n}\longrightarrow\mathbb R^{n}$ is an isometry, that $f(0)=0$.
1) Prove that for each $u,v$ we have $(u,v)=(f(u),f(v)$
2) Prove that $f$ is linear.
2012 Pre-Preparation Course Examination, 4
Prove that these two statements are equivalent for an $n$ dimensional vector space $V$:
[b]$\cdot$[/b] For the linear transformation $T:V\longrightarrow V$ there exists a base for $V$ such that the representation of $T$ in that base is an upper triangular matrix.
[b]$\cdot$[/b] There exist subspaces $\{0\}\subsetneq V_1 \subsetneq ...\subsetneq V_{n-1}\subsetneq V$ such that for all $i$, $T(V_i)\subseteq V_i$.
2010 Romania National Olympiad, 1
Let $a,b\in \mathbb{R}$ such that $b>a^2$. Find all the matrices $A\in \mathcal{M}_2(\mathbb{R})$ such that $\det(A^2-2aA+bI_2)=0$.
2007 Nicolae Coculescu, 4
Let $ n\in{N^*}$,$ n\ge{3}$ and $ a_1,a_2,...,a_n\in{R^*}$, so that $ |a_i|\neq{|a_j|}$, for every $ i,j\in{\{1,2,...,n\}}, i\neq{j}$.
Find $ p\in{S_n}$ with the property:
$ a_ia_j < \equal{} a_{p(i)}a_{p(j)}$, for every $ i,j\in{\{1,2,....n\}}$,$ i\neq{j}$ (Teodor Radu)
2004 District Olympiad, 1
Let $n\geq 2$ and $1 \leq r \leq n$. Consider the set $S_r=(A \in M_n(\mathbb{Z}_2), rankA=r)$. Compute the sum $\sum_{X \in S_r}X$
1950 Miklós Schweitzer, 4
Put
$ M\equal{}\begin{pmatrix}p&q&r\\
r&p&q\\q&r&p\end{pmatrix}$
where $ p,q,r>0$ and $ p\plus{}q\plus{}r\equal{}1$. Prove that
$ \lim_{n\rightarrow \infty}M^n\equal{}\begin{bmatrix}\frac13&\frac13&\frac13\\
\frac13&\frac13&\frac13\\\frac13&\frac13&\frac13\end{bmatrix}$
2012 Pre-Preparation Course Examination, 1
Suppose that $W,W_1$ and $W_2$ are subspaces of a vector space $V$ such that $V=W_1\oplus W_2$. Under what conditions we have
$W=(W\cap W_1)\oplus(W\cap W_2)$?
1994 Irish Math Olympiad, 4
Consider all $ m \times n$ matrices whose all entries are $ 0$ or $ 1$. Find the number of such matrices for which the number of $ 1$-s in each row and in each column is even.
2001 Romania National Olympiad, 2
We consider a matrix $A\in M_n(\textbf{C})$ with rank $r$, where $n\ge 2$ and $1\le r\le n-1$.
a) Show that there exist $B\in M_{n,r}(\textbf{C}), C\in M_{r,n}(\textbf{C})$, with $%Error. "rank" is a bad command.
B=%Error. "rank" is a bad command.
C = r$, such that $A=BC$.
b) Show that the matrix $A$ verifies a polynomial equation of degree $r+1$, with complex coefficients.