The rule gives a neat formula for solving a linear system A bit of notation first. We denote by the square matrix obtained by replacing the i-th column of Let Then CRAMER’S RULE

The proof is not too hard, but let’s do an example first.

Just trust me that Then Cramer’s rule tells us that:

Now to the proof. We have We need a simple fact of matrix algebra. Let

be matrices such that the product Let Then

A nice formula for

The equation

Your textbook calls the number

DETERMINANTS AND AREAS/VOLUMES This part of the textbook Is rather easy and is left for work in the tutorials. The important facts are theorems 9 and 10: (pp. 180 and 182 resp.) Theorem 9 says that the columns of a 2x2 or 3x3 matrix determine a parallelogram/paralleopiped whose area/volume is det A Theorem 10 says that a linear transformation with standard matrix A alters areas/volumes by a factor of det A.