## 每週問題 June 26, 2017

Let $A$ be an $n\times n$ matrix and $\hbox{rank}A=1$. Prove that $\det (A+I)=\hbox{trace}A+1$.

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## 每週問題 June 19, 2017

Prove that an invertible matrix $A$ can be represented in the form $A=QR$, where $Q$ is an orthogonal matrix and $R$ is an upper triangular matrix.

## 每週問題 June 12, 2017

Let $\mathbf{v}_1,\ldots,\mathbf{v}_n$ be a basis of an inner product space. Prove that there exists an orthogonal basis $\mathbf{e}_1,\ldots,\mathbf{e}_n$ such that $\mathbf{e}_i\in\hbox{span}\{\mathbf{v}_1,\ldots,\mathbf{v}_i\}$ for all $i=1,\ldots,n$.

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## 每週問題 June 5, 2017

Let $A$ and $B$ be $n\times n$ matrices, where $n$ is an odd number. Prove that if $AB=0$ then at least one of the matrices $A+A^T$ and $B+B^T$ is singular.

## 每週問題 May 29, 2017

Let $A$ and $B$ be complex matrices of size $m\times n$ and $p\times n$, respectively. If $N(A)\subset N(B)$, prove that $B=XA$ for some $p\times m$ matrix $X$.

## 每週問題 May 22, 2017

Suppose $A$ is $n\times n$, $B$ is $n\times 1$, $C$ is $1\times n$, and $d$ is a number. Prove that $\begin{vmatrix} A&B\\ C&d \end{vmatrix}=d|A|-C(\hbox{adj}A)B$.

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## 每週問題 May 15, 2017

Let $A$ be an $n\times n$ skew-symmetric matrix. Prove that $\hbox{adj}A$ is a symmetric matrix for odd $n$ and a skew-symmetric matrix for even $n$.

## 每週問題 May 8, 2017

Let $\mathbf{v}_1,\ldots,\mathbf{v}_{n+1}$ be vectors in $\mathbb{R}^n$ ( $n\ge 2$) such that $\mathbf{v}_i^T\mathbf{v}_j<0$ for $i\neq j$. Prove that any $n$ of these vectors form a basis of $\mathbb{R}^n$.

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## 每週問題 May 1, 2017

Let $A=[a_{ij}]$ be an $n\times n$ matrix. Prove that if $|a_{ii}|>\sum_{j\neq i}|a_{ij}|$ for $i=1,\ldots,n$, then $A$ is invertible.

Let $A$ be an $m\times n$ matrix and $B$ be an $n\times p$ matrix. Prove that $\dim (C(B)\cap N(A))=\dim C(B)-\dim C(AB)=\dim N(AB)-\dim N(B)$.
Note that $C(X)$ and $N(X)$ denote the column space and nullspace of $X$, respectively.