query: Let $A$ and $B$ be two matrices in $M_{n,p}(R)$. Show that $A^{T}A = B^{T}B \iff \exists P \in On(R)$ such that $B = PA$.
positive_passages.docid: 153290
positive_passages.text: $A^TA=B^TB$. Is $A=QB$ for some orthogonal $Q$?
negative_passages.docid: 24809
negative_passages.text: For all square matrices $A$ and $B$ of the same size, it is true that $(A+B)^2 = A^2 + 2AB + B^2$?
negative_passages.docid: 32155
negative_passages.text: Prove that if for every $x \in \mathbb{R}^N$ $Ax=Bx$ then $A=B$
negative_passages.docid: 106584
negative_passages.text: If $A^{2n}=I$, then $A^n=\pm I$ where $A\in M_n(\mathbb{R})$
negative_passages.docid: 33477
negative_passages.text: Proving that if $A$ is a $8\times 8$ matrix over $\mathbb{R}$ and $A^3=A$, then $A$ is diagonalizable.
negative_passages.docid: 1962
negative_passages.text: Polynomial $P(a)=b,P(b)=c,P(c)=a$
negative_passages.docid: 73155
negative_passages.text: Prove that for any square matrix, an invertible matrix B exists, so that BA is triangular
negative_passages.docid: 61274
negative_passages.text: If $AA^T$ is the zero matrix, then $A$ is the zero matrix
negative_passages.docid: 88746
negative_passages.text: Positive definiteness of the matrix $A+B$
negative_passages.docid: 80892
negative_passages.text: Does $\wp(A \cap B) = \wp(A) \cap \wp(B)$ hold? How to prove it?
negative_passages.docid: 30583
negative_passages.text: If $a\mid b$ and $b\mid a$, $a,b\in R$, show that $\exists w \in R^\times$ with $b=wa$
negative_passages.docid: 109745
negative_passages.text: If $(a,k)=(b,m)=1$, prove that $(ab,km)=(a,m)(b,k)$.
negative_passages.docid: 14221
negative_passages.text: How can we prove it? ${a}{b}≤\frac{a^p}{p}+\frac{b^q}{q}$
negative_passages.docid: 36117
negative_passages.text: Prove that matrices of the form $\begin{pmatrix} x & x \\ x & x \end{pmatrix}$ are a group under matrix multiplication.
negative_passages.docid: 74460
negative_passages.text: Matrices A+B=AB implies A commutes with B
negative_passages.docid: 128760
negative_passages.text: If $A$ and $B$ two $n\times n$ matrix such that $AB=BA$, then $\dim(A\cap B)$?
negative_passages.docid: 115582
negative_passages.text: Prove that if a product$ AB$ of $n\times n$ matrices is invertible, so are the factors $A$ and $B$.