Found problems: 93
2004 District Olympiad, 4
Let $A=(a_{ij})\in \mathcal{M}_p(\mathbb{C})$ such that $a_{12}=a_{23}=\ldots=a_{p-1,p}=1$ and $a_{ij}=0$ for any other entry.
a)Prove that $A^{p-1}\neq O_p$ and $A^p=O_p$.
b)If $X\in \mathcal{M}_{p}(\mathbb{C})$ and $AX=XA$, prove that there exist $a_1,a_2,\ldots,a_p\in \mathbb{C}$ such that:
\[X=\left( \begin{array}{ccccc} a_1 & a_2 & a_3 & \ldots & a_p \\ 0 & a_1 & a_2 & \ldots & a_{p-1} \\ 0 & 0 & a_1 & \ldots & a_{p-2} \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & a_1 \end{array} \right)\]
c)If there exist $B,C\in \mathcal{M}_p(\mathbb{C})$ such that $(I_p+A)^n=B^n+C^n,\ (\forall)n\in \mathbb{N}^*$, prove that $B=O_p$ or $C=O_p$.
2001 District Olympiad, 1
Let $A\in \mathcal{M}_2(\mathbb{R})$ such that $\det(A)=d\neq 0$ and $\det(A+dA^*)=0$. Prove that $\det(A-dA^*)=4$.
[i]Daniel Jinga[/i]
2006 Iran MO (3rd Round), 2
$f: \mathbb R^{n}\longrightarrow\mathbb R^{m}$ is a non-zero linear map. Prove that there is a base $\{v_{1},\dots,v_{n}m\}$ for $\mathbb R^{n}$ that the set $\{f(v_{1}),\dots,f(v_{n})\}$ is linearly independent, after ommitting Repetitive elements.
2001 IMC, 1
Let $ n$ be a positive integer. Consider an $ n\times n$ matrix with entries $ 1,2,...,n^2$ written in order, starting at the top left and moving along each row in turn left-to-right. (e.g. for $ n \equal{} 3$ we get $ \left[\begin{array}{ccc}1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9\end{array}\right]$)
We choose $ n$ entries of the matrix such that exactly one entry is chosen in each row and each column. What are the possible values of the sum of the selected entries?
2000 IMC, 3
Let $A,B\in\mathbb{C}^{n\times n}$ with $\rho(AB - BA) = 1$. Show that $(AB - BA)^2 = 0$.
1950 Miklós Schweitzer, 8
Let $ A \equal{} (a_{ik})$ be an $ n\times n$ matrix with nonnegative elements such that $ \sum_{k \equal{} 1}^n a_{ik} \equal{} 1$ for $ i \equal{} 1,...,n$.
Show that, for every eigenvalue $ \lambda$ of $ A$, either $ |\lambda| < 1$ or there exists a positive integer $ k$ such that $ \lambda^k \equal{} 1$
1997 IMC, 4
(a) Let $f: \mathbb{R}^{n\times n}\rightarrow\mathbb{R}$ be a linear mapping. Prove that $\exists ! C\in\mathbb{R}^{n\times n}$ such that $f(A)=Tr(AC), \forall A \in \mathbb{R}^{n\times n}$.
(b) Suppose in addtion that $\forall A,B \in \mathbb{R}^{n\times n}: f(AB)=f(BA)$. Prove that $\exists \lambda \in \mathbb{R}: f(A)=\lambda Tr(A)$
2006 District Olympiad, 1
Let $x>0$ be a real number and $A$ a square $2\times 2$ matrix with real entries such that $\det {(A^2+xI_2 )} = 0$.
Prove that $\det{ (A^2+A+xI_2) } = x$.
2010 IberoAmerican Olympiad For University Students, 5
Let $A,B$ be matrices of dimension $2010\times2010$ which commute and have real entries, such that $A^{2010}=B^{2010}=I$, where $I$ is the identity matrix. Prove that if $\operatorname{tr}(AB)=2010$, then $\operatorname{tr}(A)=\operatorname{tr}(B)$.
2007 District Olympiad, 4
Let $A,B\in \mathcal{M}_n(\mathbb{R})$ such that $B^2=I_n$ and $A^2=AB+I_n$. Prove that:
\[\det A\le \left(\frac{1+\sqrt{5}}{2}\right)^n\]
2003 District Olympiad, 1
In the $xOy$ system, consider the collinear points $A_i(x_i,y_i),\ 1\le i\le 4$, such that there are invertible matrices $M\in \mathcal{M}_4(\mathbb{C})$ such that $(x_1,x_2,x_3,x_4)$ and $(y_1,y_2,y_3,y_4)$ are their first two lines. Prove that the sum of the entries of $M^{-1}$ doesn't depend of $M$.
[i]Marian Andronache[/i]
2007 Nicolae Păun, 1
Prove that $ \exists X,Y,Z\in \mathcal{M}_n(\mathbb{C})$ such that
a)$ X^2\plus{}Y^2\equal{}A$
b) $ X^3\plus{}Y^3\plus{}Z^3\equal{}A$ , where $ A\in \mathcal{M}_n(\mathbb{C})$
2007 IMS, 7
$x_{1},x_{2},\dots,x_{n}$ are real number such that for each $i$, the set $\{x_{1},x_{2},\dots,x_{n}\}\backslash \{x_{i}\}$ could be partitioned into two sets that sum of elements of first set is equal to the sum of the elements of the other. Prove that all of $x_{i}$'s are zero.
[hide="Hint"]It is a number theory problem.[/hide]
1987 Romania Team Selection Test, 1
Let $a,b,c$ be distinct real numbers such that $a+b+c>0$. Let $M$ be the set of $3\times 3$ matrices with the property that each line and each column contain all given numbers $a,b,c$. Find $\{\max \{ \det A \mid A \in M \}$ and the number of matrices which realise the maximum value.
[i]Mircea Becheanu[/i]
2006 Iran MO (3rd Round), 1
Suppose that $A\in\mathcal M_{n}(\mathbb R)$ with $\text{Rank}(A)=k$. Prove that $A$ is sum of $k$ matrices $X_{1},\dots,X_{k}$ with $\text{Rank}(X_{i})=1$.
2010 District Olympiad, 2
Consider the matrix $ A,B\in \mathcal l{M}_3(\mathbb{C})$ with $ A=-^tA$ and $ B=^tB$. Prove that if the polinomial function defined by
\[ f(x)=\det(A+xB)\]
has a multiple root, then $ \det(A+B)=\det B$.
1997 IMC, 3
Let $A,B \in \mathbb{R}^{n\times n}$ with $A^2+B^2=AB$. Prove that if $BA-AB$ is invertible then $3|n$.
2012 Pre-Preparation Course Examination, 6
Suppose that $V$ is a finite dimensional vector space over the real numbers equipped with an inner product and $S:V\times V \longrightarrow \mathbb R$ is a skew symmetric function that is linear for each variable when others are kept fixed. Prove there exists a linear transformation $T:V \longrightarrow V$ such that
$\forall u,v \in V: S(u,v)=<u,T(v)>$.
We know that there always exists $v\in V$ such that $W=<v,T(v)>$ is invariant under $T$. (it means $T(W)\subseteq W$). Prove that if $W$ is invariant under $T$ then the following subspace is also invariant under $T$:
$W^{\perp}=\{v\in V:\forall u\in W <v,u>=0\}$.
Prove that if dimension of $V$ is more than $3$, then there exist a two dimensional subspace $W$ of $V$ such that the volume defined on it by function $S$ is zero!!!!
(This is the way that we can define a two dimensional volume for each subspace $V$. This can be done for volumes of higher dimensions.)
2009 District Olympiad, 1
Let $A,B,C\in \mathcal{M}_3(\mathbb{R})$ such that $\det A=\det B=\det C$ and $\det(A+iB)=\det(C+iA)$. Prove that $\det (A+B)=\det (C+A)$.
2004 Romania National Olympiad, 2
Let $n \in \mathbb N$, $n \geq 2$.
(a) Give an example of two matrices $A,B \in \mathcal M_n \left( \mathbb C \right)$ such that \[ \textrm{rank} \left( AB \right) - \textrm{rank} \left( BA \right) = \left\lfloor \frac{n}{2} \right\rfloor . \]
(b) Prove that for all matrices $X,Y \in \mathcal M_n \left( \mathbb C \right)$ we have \[ \textrm{rank} \left( XY \right) - \textrm{rank} \left( YX \right) \leq \left\lfloor \frac{n}{2} \right\rfloor . \]
[i]Ion Savu[/i]
2005 Olympic Revenge, 4
Let A be a symmetric matrix such that the sum of elements of any row is zero.
Show that all elements in the main diagonal of cofator matrix of A are equal.
2002 District Olympiad, 3
a)Find a matrix $A\in \mathcal{M}_3(\mathbb{C})$ such that $A^2\neq O_3$ and $A^3=O_3$.
b)Let $n,p\in\{2,3\}$. Prove that if there is bijective function $f:\mathcal{M}_n(\mathbb{C})\rightarrow \mathcal{M}_p(\mathbb{C})$ such that $f(XY)=f(X)f(Y),\ \forall X,Y\in \mathcal{M}_n(\mathbb{C})$, then $n=p$.
[i]Ion Savu[/i]
1997 IMC, 2
Let $M \in GL_{2n}(K)$, represented in block form as \[ M = \left[ \begin{array}{cc} A & B \\ C & D \end{array} \right] , M^{-1} = \left[ \begin{array}{cc} E & F \\ G & H \end{array} \right] \]
Show that $\det M.\det H=\det A$.
1998 Iran MO (3rd Round), 3
Let $A,B$ be two matrices with positive integer entries such that sum of entries of a row in $A$ is equal to sum of entries of the same row in $B$ and sum of entries of a column in $A$ is equal to sum of entries of the same column in $B$. Show that there exists a sequence of matrices $A_1,A_2,A_3,\cdots , A_n$ such that all entries of the matrix $A_i$ are positive integers and in the sequence
\[A=A_0,A_1,A_2,A_3,\cdots , A_n=B,\]
for each index $i$, there exist indexes $k,j,m,n$ such that
\[\begin{array}{*{20}{c}}
\\
{{A_{i + 1}} - {A_{i}} = }
\end{array}\begin{array}{*{20}{c}}
{\begin{array}{*{20}{c}}
\quad \quad \ \ j& \ \ \ {k}
\end{array}} \\
{\begin{array}{*{20}{c}}
m \\
n
\end{array}\left( {\begin{array}{*{20}{c}}
{ + 1}&{ - 1} \\
{ - 1}&{ + 1}
\end{array}} \right)}
\end{array} \ \text{or} \ \begin{array}{*{20}{c}}
{\begin{array}{*{20}{c}}
\quad \quad \ \ j& \ \ \ {k}
\end{array}} \\
{\begin{array}{*{20}{c}}
m \\
n
\end{array}\left( {\begin{array}{*{20}{c}}
{ - 1}&{ + 1} \\
{ + 1}&{ - 1}
\end{array}} \right)}
\end{array}.\]
That is, all indices of ${A_{i + 1}} - {A_{i}}$ are zero, except the indices $(m,j), (m,k), (n,j)$, and $(n,k)$.
2005 District Olympiad, 3
a)Let $A,B\in \mathcal{M}_3(\mathbb{R})$ such that $\text{rank}\ A>\text{rank}\ B$. Prove that $\text{rank}\ A^2\ge \text{rank}\ B^2$.
b)Find the non-constant polynomials $f\in \mathbb{R}[X]$ such that $(\forall)A,B\in \mathcal{M}_4(\mathbb{R})$ with $\text{rank}\ A>\text{rank}\ B$, we have $\text{rank}\ f(A)>\text{rank}\ f(B)$.