1 분 소요

# Questions

## Question 1

Show that for any square matrix, the matrix $A + \mathbf{A}^\top$ is symmetric.

## Question 2

If $A = (a_{ij})$ is a square matrix, then the elements $a_{ii}$ are called the diagonal elements. How do the diagonal elements of $A$ and $\mathbf{A}^\top$ differ?

## Question 3

Define a matrix $A$ to be skew-symmetric if $\mathbf{A}^\top = -A$. Show that for any square matrix $A$, the matrix $A-\mathbf{A}^\top$ is skew-symmetric.

## Question 4

If a matrix is skew-symmetric, what can you say about its diagonal elements?

the matrix $A + \mathbf{A}^\top$ is symmetric.

$\because (A + \mathbf{A}^\top)^\top = (\mathbf{A}^\top + A)$ $= (A+\mathbf{A}^\top)$

Same

$A_{i,j} = \begin{pmatrix} \color{green}{a_{1,1}} & a_{1,2} & \cdots & a_{1,j} \\ a_{2,1} & \color{green}{a_{2,2}} & \cdots & a_{2,j} \\ \vdots & \vdots & \ddots & \vdots \\ a_{i,1} & a_{i,2} & \cdots & \color{green}{a_{i,j}} \end{pmatrix}$ $\mathbf{(A)}^\top_{i,j} = \begin{pmatrix} \color{red}{a_{1,1}} & a_{2,1} & \cdots & a_{i,1} \\ a_{1,2} & \color{red}{a_{2,2}} & \cdots & a_{i,2} \\ \vdots & \vdots & \ddots & \vdots \\ a_{1,j} & a_{2,j} & \cdots & \color{red}{a_{i,j}} \end{pmatrix}$

The matrix $A - \mathbf{A}^\top$ is a skew-symmetric.

$\because \mathbf{(A - \mathbf{A}^\top)}^\top = \mathbf{A}^\top - A$ $= -(A - \mathbf{A}^\top)$

$\text{By skew-symmetric}$ $a_{ii} = -a_{ii}$ $\therefore a_{ii} = 0$