I have two problems I've been assigned to assist in studying for an exam, but I can't figure out how to proceed. The first one is:
Prove the following:
|1 a bc| --|1 a a^2|----------------- |1 a a^2|
|1 b ac| = |1 b b^2| = (c-a)(b-a)(c-b)|0 1 b+a |=(c-a)(b-a)(c-b)
|1 c ab| --|1 c c^2|----------------- |0 0 1 |
Using only these rules:
1: If each element of a one row or column of a determinant is multiplied by a number k, the value of the determinant is multiplied by k.
2: The value of a determinant is zero if
a: all elements of one row or colum are zero, or if
b: two rows or columns are identical, or if
c: two rows or colums are proportional
3: If two rows or colums of a determinant are interchanged, the value of the determinant changes sign.
4: The value of a determinant is unchanged if
a: rows are written as columns and columns as rows, or if
b: we add to each element of one row or column, k times the corresponding element of another row or column, where k is any number.
My second question is more general.
When trying to determine the determinant of a matrix containing polynomials by hand, is there any way to do it other than brute forcing it with tons of polynomial multiplication?