9. Determinant of transposed matrix

The transposed matrix A' is obtained by interchange of the row and columns of the matrix A:

The rows of A' are the columns of A, the columns of A' are the rows of A.

As is readily seen from (6.1), each term in the sum involves an element of each column of A, whence |A| is linear and homogeneous with respect to the elements of each column of A. One can therefore say:

|A| is linear and homogenous with respect to the elements of each row of A'.

We will now examine the changes in |A| as two columns of A or, what is the same, two rows of A' are interchanged. Denote by A* the matrix A after interchange of two given columns. If you interchange two given numbers, a permutation P becomes a well defined permutation P*. If you denote corresponding to the permutation

P = [p₁, ··· ,p_n]

the term

by

c(P)·F_P,

you can rewrite (6.1)

Interchange of two columns of A converts it into

Since P covers simultaneously with P all permutations, you can also write

because this sum contains the same terms as (9.1), but in a different order. By Theorem 2,

c(P^*) = - c(P).

A comparison of (9.2) with (9.3) yields now

|A^*| = -|A|,

whence we can say:

|A| changes its sign when you interchange two rows of A'.

Since, obviously, you find for the unit matrix E' = E

|E'| = 1.

Now imagine |A| to be a polynomial in the elements of A'. We have discovered that this polynomial has the properties, demanded in Theorem 3 and therefore agrees with the determinant, whence follows

Theorem 4

The determinant |A| does not change as you interchange rows and columns:

|A| = |A'|.

Consequently, all results regarding determinants remain valid if one interchanges the word "row" and "column" and conversely.

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