## Cramer's rule and invertibility criterion

### Consequences of multilinearity

For a fixed is a linear function of column Such a linear function generates a row-vector by way of a formula (see Exercise 3)

(1)

**Exercise 1**. In addition to (1), we have

(2) for any

**Proof**. Here and in the future it is useful to introduce the coordinate representation for and put

(3)

Then we can write (1) as Here the element does not involve and therefore by the different-columns-different-rows rule it does not involve elements of the entire column Hence, 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.

### Cramer's rule

**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

(4)

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.

### Invertibility criterion

**Exercise 3**. is invertible if and only if

**Proof**. If is invertible, then By multiplicativity of determinant and Axiom 3 this implies Thus,

Conversely, suppose (1), (2) and (3) imply

(5)

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.

**Definition 1**. The matrix is more than a transient technical twist; it is called an **adjugate matrix** and property (5), correspondingly, is called an **adjugate identity**.