What is $Var[b]$ in multiple regression?

Assume a linear regression model $y=X\beta +\u03f5$ with $\u03f5\sim N(0,{\sigma}^{2}I)$ and $\hat{y}=Xb$ where $b=({X}^{\prime}X{)}^{-1}{X}^{\prime}y$. Besides $H=X({X}^{\prime}X{)}^{-1}{X}^{\prime}$ is the linear projection from the response space to the span of $X$, i.e., $\hat{y}=Hy$

Now I want to calculate $Var[b]$ but what I get is an $k\times k$ matrix, not an $n\times n$ one. Here's my calculation:

$\begin{array}{rl}Var[b]=& \phantom{\rule{thickmathspace}{0ex}}Var[({X}^{\prime}X{)}^{-1}{X}^{\prime}y]\\ =& \phantom{\rule{thickmathspace}{0ex}}({X}^{\prime}X{)}^{-1}{X}^{\prime}\phantom{\rule{thinmathspace}{0ex}}\underset{={\sigma}^{2}I}{\underset{\u23df}{Var[y]}}X({X}^{\prime}X{)}^{-1}\\ \\ \text{Here you can}& \text{see already this thing will be k}\times \text{k}\\ \\ =& \phantom{\rule{thickmathspace}{0ex}}{\sigma}^{2}\underset{I}{\underset{\u23df}{({X}^{\prime}X{)}^{-1}{X}^{\prime}X}}({X}^{\prime}X{)}^{-1}\\ =& {\sigma}^{2}({X}^{\prime}X{)}^{-1}\phantom{\rule{thinmathspace}{0ex}}\in {R}^{k\times k}\end{array}$

What am I doing wrong?

Besides, are $E[b]=\beta $, $E[\hat{y}]=HX\beta $, $Var[\hat{y}]={\sigma}^{2}H$, $E[y-\hat{y}]=(I-H)X\beta $, $Var[y-\hat{y}]=(I-H){\sigma}^{2}$ correct (this is just on a side note, my main question is the one above)?

Assume a linear regression model $y=X\beta +\u03f5$ with $\u03f5\sim N(0,{\sigma}^{2}I)$ and $\hat{y}=Xb$ where $b=({X}^{\prime}X{)}^{-1}{X}^{\prime}y$. Besides $H=X({X}^{\prime}X{)}^{-1}{X}^{\prime}$ is the linear projection from the response space to the span of $X$, i.e., $\hat{y}=Hy$

Now I want to calculate $Var[b]$ but what I get is an $k\times k$ matrix, not an $n\times n$ one. Here's my calculation:

$\begin{array}{rl}Var[b]=& \phantom{\rule{thickmathspace}{0ex}}Var[({X}^{\prime}X{)}^{-1}{X}^{\prime}y]\\ =& \phantom{\rule{thickmathspace}{0ex}}({X}^{\prime}X{)}^{-1}{X}^{\prime}\phantom{\rule{thinmathspace}{0ex}}\underset{={\sigma}^{2}I}{\underset{\u23df}{Var[y]}}X({X}^{\prime}X{)}^{-1}\\ \\ \text{Here you can}& \text{see already this thing will be k}\times \text{k}\\ \\ =& \phantom{\rule{thickmathspace}{0ex}}{\sigma}^{2}\underset{I}{\underset{\u23df}{({X}^{\prime}X{)}^{-1}{X}^{\prime}X}}({X}^{\prime}X{)}^{-1}\\ =& {\sigma}^{2}({X}^{\prime}X{)}^{-1}\phantom{\rule{thinmathspace}{0ex}}\in {R}^{k\times k}\end{array}$

What am I doing wrong?

Besides, are $E[b]=\beta $, $E[\hat{y}]=HX\beta $, $Var[\hat{y}]={\sigma}^{2}H$, $E[y-\hat{y}]=(I-H)X\beta $, $Var[y-\hat{y}]=(I-H){\sigma}^{2}$ correct (this is just on a side note, my main question is the one above)?