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Determinant problem, matrices wee!

  1. Oct 28, 2005 #1
    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 = ?
    I made it Transposed so its more readable...really it is just
    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. jcsd
  3. Oct 28, 2005 #2

    TD

    User Avatar
    Homework Helper

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

    [tex]\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|[/tex]

    By the way, for b: mind that every single row-swap changes the sign, so an even number of swaps...
     
  4. Oct 29, 2005 #3
    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!
     
  5. Oct 29, 2005 #4

    TD

    User Avatar
    Homework Helper

    Indeed :wink:
     
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