What is the value of the determinant of a 3x3 matrix and how do I expand it?

Expand a 3x3 determinant along the first row using cofactors. For rows (a, b, c), (d, e, f), (g, h, i), the value is a times (ei − fh) minus b times (di − fg) plus c times (dh − eg). The crucial part is the alternating signs: plus, minus, plus across the top row. Each bracket is the 2x2 determinant left after deleting the row and column of that entry, called a minor. So for a you delete its row and column and are left with rows (e, f) and (h, i). Be very careful with the minus sign on the middle term, that is where most mistakes happen. You can also expand along any row or column using the same sign pattern; choosing one with zeros makes the work shorter.