You can use a single shape to create an Illustrator checkerboard and combine that with multiple transforms via the effect menu and distort and transform and then use the distort command.
I would suggest using a very basic shape to start, something like a square of 30 by 30. Go to the transform command and then enter the number of copies for the checkerboard to be 1 and set the move to be 30 and 30 (the size of the shape). You will now have two squares, one shifted to the right and down. This transform can be edited at any point via the appearance panel by double clicking the transform entry.
Go to the distort and transform again and use another transform, Illustrator will mention that you are applying a new transform (yes, that is what we want) so please ignore the warning. Enter the horizontal move value to be 60 or twice the size of the shape's width (if you change the shape, you can always update this value and unfortunately there is no hidden mathematical model for the width and height in Illustrator (or is there ?). Set the number of copies to 20 say.
Apply the transform again to this design and set the value for the vertical this time to 60 or twice the shape's height and again all the values can be updated at a later date. You now have your design based on the single shape.
You can re-size, rotate, change the points, etc and all the changes will be reflected in the rest of the design. You can also save the design as a graphic style via the graphic style panel and the right side menu and make a new graphic style. You can then apply this graphic style in the future to all other kinds of shapes such as circles, stars, type etc. You can also change the color. You can add gradients to the design and more.
You can vary all the transforms such as add in scaling into the mix such as 98% re-size for the artwork. You can also change the number of columns and rows.