Question

Does every matrix have a determinant?

Answer 1

Every SQUARE matrix `nxxn` has a determinant.
The determinant `|A|` of a square matrix `A` is a number that helps you to decide:

1) What kind of solutions a system (from whose coefficients you built the square matrix `A`) can have (unique, no solutions or an infinite number of solutions);

2) If your matrix `A`, considered as an operator that produce transformations on vectors (making them bigger, flipping them, reducing them...etc.), can have an inverse (operating an inverse transformation) and what is the size of the transformation produced by `A`.

You can have a look to System of Linear Equations and Eigenvalues/Eigenvectors.