query: Let $A$ and $x$ be $n \times n$ and $n \times 1$ matrices, all entries real and strictly positive. Assume that $A^2 x = x$. Show that $A x = x$.
positive_passages.docid: 154327
positive_passages.text: $A^2X = X$ show that $AX = X$, if $A_{ij} > 0$ and $x_{i} > 0$
negative_passages.docid: 128990
negative_passages.text: Prove that, at least one of the matrices $A+B$ and $A-B$ has to be singular
negative_passages.docid: 118417
negative_passages.text: Is $AA^T$ positive semidefinite?
negative_passages.docid: 64747
negative_passages.text: If $A^m=I$ then A is Diagonalizable
negative_passages.docid: 61274
negative_passages.text: If $AA^T$ is the zero matrix, then $A$ is the zero matrix
negative_passages.docid: 6960
negative_passages.text: Necessary and sufficient condition for the matrix $A = I - 2 x x^t$ to be orthogonal
negative_passages.docid: 89729
negative_passages.text: If $X$ is a singular matrix, then is $X'X$ singular?
negative_passages.docid: 142360
negative_passages.text: How to show $x_1,x_2, \dots ,x_n \geq 0 $ and $ x_1 + x_2 + \dots + x_n \leq \frac{1}{2} \implies (1-x_1)(1-x_2) \cdots (1-x_n) \geq \frac{1}{2}$
negative_passages.docid: 138025
negative_passages.text: How to show the solution of $\dot{x} = Ax $ is an invariant subspace?
negative_passages.docid: 42292
negative_passages.text: Show that $a_n<\sqrt{2}$ for every $n\in\mathbb{N}$.
negative_passages.docid: 41411
negative_passages.text: How to show that the nth power of a $n \times n$ nilpotent matrix equals to zero $A^n=0$
negative_passages.docid: 87425
negative_passages.text: Prove that if $A$ is a square matrix with integer entries and $\det(A) = \pm 1$, then $A^{-1}$ contains all integer entries.
negative_passages.docid: 126711
negative_passages.text: What is the square matrix $A$ equal to if $A=A^2$
negative_passages.docid: 55031
negative_passages.text: Show any matrix $A_{n\times n}$ can be written as sum of two nonsingular matrices
negative_passages.docid: 83337
negative_passages.text: If $A ∈M_n(C)$ is such that $(Ax,x) = 0$ for all $x ∈C^n$, then $A = 0$
negative_passages.docid: 89792
negative_passages.text: Let $x,y$ be positive real numbers. Then $x<y$ iff $x^n<y^n$ for all $n\in \mathbb{N}$
negative_passages.docid: 36314
negative_passages.text: If $Ax = Bx$ for all $x \in C^{n}$, then $A = B$.