Found problems: 93
2008 District Olympiad, 2
Let $A,B\in \mathcal{M}_n(\mathbb{R})$. Prove that $\text{rank}\ A+\text{rank}\ B\le n$ if and only if there exists an invertible matrix $X\in \mathcal{M}_n(\mathbb{R})$ such that $AXB=O_n$.
2003 IMC, 3
Let $A\in\mathbb{R}^{n\times n}$ such that $3A^3=A^2+A+I$. Show that the sequence $A^k$ converges to an idempotent matrix. (idempotent: $B^2=B$)
1975 Miklós Schweitzer, 10
Prove that an idempotent linear operator of a Hilbert space is self-adjoint if and only if it has norm $ 0$ or $ 1$.
[i]J. Szucs[/i]
1983 Iran MO (2nd round), 3
Find a matrix $A_{(2 \times 2)}$ for which
\[ \begin{bmatrix}2 &1 \\ 3 & 2\end{bmatrix} A \begin{bmatrix}3 & 2 \\ 4 & 3\end{bmatrix} = \begin{bmatrix}1 & 2 \\ 2 & 1\end{bmatrix}.\]
2001 District Olympiad, 2
Let $n\in \mathbb{N},\ n\ge 2$. For any matrix $A\in \mathcal{M}_n(\mathbb{C})$, let $m(A)$ be the number of non-zero minors of $A$. Prove that:
a)$m(I_n)=2^n-1$;
b)If $A\in \mathcal{M}_n(\mathbb{C})$ is non-singular, then $m(A)\ge 2^n-1$.
[i]Marius Ghergu[/i]
1998 IMC, 1
Let $V$ be a 10-dimensional real vector space and $U_1,U_2$ two linear subspaces such that $U_1 \subseteq U_2, \dim U_1 =3, \dim U_2=6$. Let $\varepsilon$ be the set of all linear maps $T: V\rightarrow V$ which have $T(U_1)\subseteq U_1, T(U_2)\subseteq U_2$. Calculate the dimension of $\varepsilon$. (again, all as real vector spaces)
2009 Romania National Olympiad, 3
Let $A,B\in \mathcal{M}_n(\mathbb{C})$ such that $AB=BA$ and $\det B\neq 0$.
a) If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$.
b) Is the question from a) still true if $AB\neq BA$ ?
2005 Brazil Undergrad MO, 1
Determine the number of possible values for the determinant of $A$, given that $A$ is a $n\times n$ matrix with real entries such that $A^3 - A^2 - 3A + 2I = 0$, where $I$ is the identity and $0$ is the all-zero matrix.
2013 Romania National Olympiad, 2
Whether $m$ and $n$ natural numbers, $m,n\ge 2$. Consider matrices, ${{A}_{1}},{{A}_{2}},...,{{A}_{m}}\in {{M}_{n}}(R)$ not all nilpotent. Demonstrate that there is an integer number $k>0$ such that ${{A}^{k}}_{1}+{{A}^{k}}_{2}+.....+{{A}^{k}}_{m}\ne {{O}_{n}}$
2006 Iran MO (3rd Round), 4
$f: \mathbb R^{n}\longrightarrow\mathbb R^{n}$ is a bijective map, that Image of every $n-1$-dimensional affine space is a $n-1$-dimensional affine space.
1) Prove that Image of every line is a line.
2) Prove that $f$ is an affine map. (i.e. $f=goh$ that $g$ is a translation and $h$ is a linear map.)
2012 Pre-Preparation Course Examination, 5
Suppose that for the linear transformation $T:V \longrightarrow V$ where $V$ is a vector space, there is no trivial subspace $W\subset V$ such that $T(W)\subseteq W$. Prove that for every polynomial $p(x)$, the transformation $p(T)$ is invertible or zero.
2007 Romania National Olympiad, 1
Let $A,B\in\mathcal{M}_{2}(\mathbb{R})$ (real $2\times 2$ matrices), that satisfy $A^{2}+B^{2}=AB$. Prove that $(AB-BA)^{2}=O_{2}$.
2007 District Olympiad, 2
Let $A\in \mathcal{M}_n(\mathbb{R}^*)$. If $A\cdot\ ^t A=I_n$, prove that:
a)$|\text{Tr}(A)|\le n$;
b)If $n$ is odd, then $\det(A^2-I_n)=0$.
2006 District Olympiad, 2
Let $n,p \geq 2$ be two integers and $A$ an $n\times n$ matrix with real elements such that $A^{p+1} = A$.
a) Prove that $\textrm{rank} \left( A \right) + \textrm{rank} \left( I_n - A^p \right) = n$.
b) Prove that if $p$ is prime then \[ \textrm{rank} \left( I_n - A \right) = \textrm{rank} \left( I_n - A^2 \right) = \ldots = \textrm{rank} \left( I_n - A^{p-1} \right) . \]
2009 IberoAmerican Olympiad For University Students, 2
Let $x_1,\cdots, x_n$ be nonzero vectors of a vector space $V$ and $\varphi:V\to V$ be a linear transformation such that $\varphi x_1 = x_1$, $\varphi x_k = x_k - x_{k-1}$ for $k = 2, 3,\ldots,n$.
Prove that the vectors $x_1,\ldots,x_n$ are linearly independent.
1979 IMO Longlists, 7
$M = (a_{i,j} ), \ i, j = 1, 2, 3, 4$, is a square matrix of order four. Given that:
[list]
[*] [b](i)[/b] for each $i = 1, 2, 3,4$ and for each $k = 5, 6, 7$,
\[a_{i,k} = a_{i,k-4};\]\[P_i = a_{1,}i + a_{2,i+1} + a_{3,i+2} + a_{4,i+3};\]\[S_i = a_{4,i }+ a_{3,i+1} + a_{2,i+2} + a_{1,i+3};\]\[L_i = a_{i,1} + a_{i,2} + a_{i,3} + a_{i,4};\]\[C_i = a_{1,i} + a_{2,i} + a_{3,i} + a_{4,i},\]
[*][b](ii)[/b] for each $i, j = 1, 2, 3, 4$, $P_i = P_j , S_i = S_j , L_i = L_j , C_i = C_j$, and
[*][b](iii)[/b] $a_{1,1} = 0, a_{1,2} = 7, a_{2,1} = 11, a_{2,3} = 2$, and $a_{3,3} = 15$.[/list]
find the matrix M.
2008 Romania National Olympiad, 4
Let $ A\equal{}(a_{ij})_{1\leq i,j\leq n}$ be a real $ n\times n$ matrix, such that $ a_{ij} \plus{} a_{ji} \equal{} 0$, for all $ i,j$. Prove that for all non-negative real numbers $ x,y$ we have \[ \det(A\plus{}xI_n)\cdot \det(A\plus{}yI_n) \geq \det (A\plus{}\sqrt{xy}I_n)^2.\]
2011 IMC, 2
Does there exist a real $3\times 3$ matrix $A$ such that $\text{tr}(A)=0$ and $A^2+A^t=I?$ ($\text{tr}(A)$ denotes the trace of $A,\ A^t$ the transpose of $A,$ and $I$ is the identity matrix.)
[i]Proposed by Moubinool Omarjee, Paris[/i]