query: Show that rank($A^{n+1}$) = rank($A^n$)
positive_passages.docid: 146847
positive_passages.text: Given a square matrix A of order n, prove $\operatorname{rank}(A^n) = \operatorname{rank}(A^{n+1})$
negative_passages.docid: 119436
negative_passages.text: Proving $2^n\leq 2^{n+1}-2^{n-1}-1$ for all $n\geq 1$ by induction
negative_passages.docid: 101286
negative_passages.text: How show that $|a_{n}-1|\le c\lambda ^n,\lambda\in (0,1)$
negative_passages.docid: 97469
negative_passages.text: How to prove $|a^k|=n/\gcd(n,k)$ whenever $|a|=n$?
negative_passages.docid: 74611
negative_passages.text: $\operatorname{Adj} (\mathbf I_n x-\mathbf A)$ when $\operatorname{rank}(\mathbf A)\le n-2$
negative_passages.docid: 67058
negative_passages.text: Prove that $\text{rank}(X^TX)=\text{rank}(X)$
negative_passages.docid: 119435
negative_passages.text: Rank of the difference of matrices
negative_passages.docid: 64517
negative_passages.text: Prove that row rank of a matrix equals column rank
negative_passages.docid: 109760
negative_passages.text: Showing that the product of all $n$th Roots of Unity is $(-1)^{n+1}$
negative_passages.docid: 48081
negative_passages.text: Prove that for all $n \in \mathbb{N}$: $2^{n-1}(a^n + b^n) \ge (a + b)^n$
negative_passages.docid: 42780
negative_passages.text: Show that if $a^{n-1} \equiv 1 \pmod{n}$ for all $a$ such that $\gcd(n,a) = 1$, then $a^{n} \equiv a \pmod{n}$ for all $a$.
negative_passages.docid: 97487
negative_passages.text: How to prove whether a matrix has rank $1$
negative_passages.docid: 134402
negative_passages.text: $\mathrm{rank}(A)+\mathrm{rank}(I-A)=n$ for $A$ idempotent matrix
negative_passages.docid: 115866
negative_passages.text: Proof by induction; $a^n$ divides $b^n$ implies $a$ divides $b$
negative_passages.docid: 43000
negative_passages.text: Show $\operatorname{rank}(A) + \operatorname{rank}(B) \ge \operatorname{rank}(A+B)$
negative_passages.docid: 62254
negative_passages.text: Show $F_{n+1} \cdot F_{n-1} = F_n^2 + (-1)^n$ for all $n \in \mathbb{N}$
negative_passages.docid: 75373
negative_passages.text: Show that $2^n>n$ for $n\in \mathbb{N}$