## 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

$\begin{vmatrix} A&B\\ C&d \end{vmatrix}=d|A|-C(\hbox{adj}A)B$.

## 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 $\mathbb{R}^n$.

## 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.

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## 每週問題 April 24, 2017

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.

## 每週問題 April 17, 2017

Let $A$ and $B$ be $n\times n$ matrices. If $A^2B+BA^2=2ABA$, show that $(AB-BA)^n=0$.

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## 每週問題 April 10, 2017

Let $\mathbf{x}$ and $\mathbf{y}$ be $n$-dimensional column vectors. Prove that

$\hbox{adj}(I-\mathbf{x}\mathbf{y}^T)=\mathbf{x}\mathbf{y}^T+(1-\mathbf{y}^T\mathbf{x})I$.

## 每週問題 April 3, 2017

Find all matrices commuting with $E$, where $E$ is the matrix all elements of which are equal to $1$.

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## 每週問題 March 27, 2017

Prove the following statements.
(a) Let $D=\hbox{diag}(d_1,\ldots,d_n)$, where $d_i$ are distinct. If $AD=DA$, then $A$ is a diagonal matrix.
(b) Let $D=\hbox{diag}(d_1,\ldots,d_n)$, where $d_i$ are distinct and nonzero, and $N=[n_{ij}]$ be an $n\times n$ matrix, where $n_{ij}=\delta_{i+1,j}$. If $AD=DA$ and $NDA=AND$, then $A=aI$.

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## 每週問題 March 20, 2017

Let $A$ and $B$ be normal matrices such that $C(A)\perp C(B)$, where $C(X)$ denotes the column space of $X$. Prove that $A+B$ is a normal matrix. Note that $P$ is a normal matrix if $P^\ast P=PP^\ast$.