What is an example of perfect multicollinearity? What is an example of perfect collinearity in terms of the design matrix $X$?
I would like an example where $\hat \beta =  (X'X)^{-1}X'Y$ can't be estimated because $(X'X)$ is not invertible.
 A: Some trivial examples to help intuition:

*

*$\mathbf{x_1}$ is height in centimeters. $\mathbf{x_2}$ is height in meters. Then:

*

*$\mathbf{x_1} = 100 \mathbf{x_2}$, and your design matrix $X$ will not have linearly independent columns.



*$\mathbf{x_1} = \mathbf{1}$ (i.e. you include a constant in your regression), $\mathbf{x_2}$ is temperature in fahrenheit, and $\mathbf{x_3}$ is temperature in celsius. Then:

*

*$\mathbf{x_2} = \frac{9}{5}\mathbf{x_3} + 32 \mathbf{x_1}$, and your design matrix $X$ will not have linearly independent columns.



*Everyone starts school at age 5, $\mathbf{x_1} = \mathbf{1}$ (i.e. constant value of 1 across all observations), $\mathbf{x_2}$ is years of schooling, $\mathbf{x_3}$ is age, and no one has left school. Then:

*

*$\mathbf{x_2} = \mathbf{x_3} - 5\mathbf{x_1}$, and your design matrix $X$ will not have linearly independent columns.



There a multitude of ways such that one column of data will be a linear function of your other data. Some of them are obvious (eg. meters vs. centimeters) while others can be more subtle (eg. age and years of schooling for younger children).
Notational notes:
Let $\mathbf{x_1}$ denote the first column of $X$, $\mathbf{x_2}$ the second column etc..., and  $\mathbf{1}$ denotes a vector of ones, which is what's included in the design matrix X if you include a constant in your regression.
