Determinant | IA Groups

Definition 3.36 (Determinant)

Consider an matrix with columns given by

The determinant of is defined by

where is the sign of the permutation . We can also write

Determinants in low dimensions

In with , let . Then

with and . Therefore, if , then .

In , one seeks a matrix and a scalar such that . Under the action of , volumes are scaled by the factor

the scalar triple product of the columns. General determinants expand in terms of determinants:

Proposition 3.37 (Properties of the Determinant)

The determinant is multilinear in the columns of the matrix. In particular,

for any scalar and any matrix .
The determinant is totally antisymmetric in the columns of the matrix. In particular, if we exchange two columns of , then the determinant changes sign.
for the identity matrix of any size .
If two rows or two columns of are equal, then .
If two rows or two columns of are linearly dependent, then .
if and only if the columns of are linearly independent.

As a consequence, under a column operation for some , the determinant is unchanged.
Hence, all properties above also hold for rows.
For any two matrices and ,

In particular, if is invertible, then
If is orthogonal, then .
If is unitary, then .

Property (5). Suppose for some and scalar . Define by

By multilinearity in the columns, . But the th column of is all zeros, so .

Property (7). A single term satisfies

for every permutation . Taking , and using ,

Property (8). Swapping columns an even or odd number of times introduces a factor of , so

If two indices satisfy for some , then by property (4). Otherwise there is a permutation with for all , and then . Therefore

In particular, if is invertible, then .

Property (9). If is orthogonal, then , and thus , giving .

Property (10). If is unitary, then , and thus , giving .