Cramer's rule and invertibility criterion
Consequences of multilinearity
Exercise 1. In addition to (1), we have
(2) for any
Proof. Here and for the future it is useful to introduce the coordinate representation for and put
Then we can write (1) as From the Leibniz formula one can see that here the element does not involve (the different-columns-different-rows rule). Therefore the vector does not involve elements of the column
Let denote the matrix obtained from by replacing column with column The vector for the matrix is the same as for because both vectors depend on the elements from columns other than the column numbered Since contains linearly dependent (actually two identical) columns, Using in (1) instead of we get as required.
After reading the next two sections, come back and read this statement again to appreciate its power and originality.
Exercise 2. Suppose For any denote the matrix formed by replacing the -th column of by the column vector . Then the solution of the system exists and the components of are given by
Proof. Premultiply by
Here we applied (1) and (2) (the -th component of the vector is and all others are zeros). From (4) it follows that On the other hand, from (1) we have (the vector for is the same as for see the proof of Exercise 1). The last two equations prove the statement.
Exercise 3. is invertible if and only if
Conversely, suppose (1), (2) and (3) imply
This means that the matrix is the inverse of Recall that existence of the left inverse implies that of the right inverse, so is invertible.