Determinant problem, matrices wee!

1. Oct 28, 2005

mr_coffee

Determinant problem, matrices!! wee!

Hello everyone...
I got part a, and b, and i'm stuck on c...
Suppose that a 4 x 4 matrix A with rows v_1, v_2, v_3, and v_4 has determinant det A = -6. Find the following determinants determinants:

det[v_1 v_2 v_3 v_4 + 7*v_2]^T = ?
determinant of
v_1
v_2
v_3
v_4 + 7*v_2

I tried 7*-6 = -42 which was wrong, because if u multiply a column by a constant, it just mutlipies the matrix by that constant, but i don't know what happens if u multip[ly a constant to a a row, and then add it to another row..
Any ideas?
If ur confused on what i'm talking about, here is an answer to part a:
5*v_1
v_2
v_3
v_4

det of that matrix is: 5*-6 = -30;

and part b:
v_4
v_3
v_2
v_1
det of that matrix is 6, because if u swap rows, it will change the sign of the detemrinant.

2. Oct 28, 2005

TD

You can use the (multi)linearity of the determinant:

$$\left| {\begin{array}{*{20}c} {a_{11} } & {a_{12} } \\ {\alpha a_{21} + \beta a_{21} ^\prime } & {\alpha a_{22} + \beta a_{22} ^\prime } \\ \end{array}} \right| = \alpha \left| {\begin{array}{*{20}c} {a_{11} } & {a_{12} } \\ {a_{21} } & {a_{22} } \\ \end{array}} \right| + \beta \left| {\begin{array}{*{20}c} {a_{11} } & {a_{12} } \\ {a_{21} ^\prime } & {a_{22} ^\prime } \\ \end{array}} \right|$$

By the way, for b: mind that every single row-swap changes the sign, so an even number of swaps...

3. Oct 29, 2005

mr_coffee

thank u TD! but when u said for part b...if its a even number of swaps, wouldn't the determinatant not be changedf at all? it would go from -6 to 6 to -6 to 6, oh wait yah it would thanks!

4. Oct 29, 2005

Indeed