•Matrices which are not square cannot be invertible. The reason is thatifAis anm×nmatrix andAB=IthenTAis surjective andTBisinjective, both of which requirem≤n. IfBA=Ialso, then likewisen≤m.•IfAandBaren×nmatrices withAB=I, thenBA=Ialso.Linear algebra chapter 3.•How to calculate the determinant of a square matrix, either by rowreduction, by expanding along a row or column, or as a sum overpermutations (at least for 2×2 and 3×3).•WhenAis a square matrix,Ais invertible if and only if det(A)6= 0.•The geometric meaning of determinants: ifAis ann×nmatrix andUis a region inRnwith finite volume, then vol(TA(U)) =|det(A)|vol(U);and det(A)>0 whenApreserves orientation.Linear algebra chapter 4.•Definition of a vector space. Main examples:Rn, subsets of a vectorspace, and some function spaces.•Definition of a subspace of a vector space. Examples: span of a subset;