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Math Help - These Determinants are driving me crazy! (Matrices)

  1. #1
    Junior Member JoeyCC's Avatar
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    These Determinants are driving me crazy! (Matrices)

    Hey guys,

    Firstly, Happy Thanks Giving for all of you who are Canadian! I've got a few problems with determinants which I can't solve! I'm hoping any of you can help me:





    I know the answer to the last problem of the first question is -23. Can anyone show me the rest? I thank you in advanced!
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  2. #2
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    Quote Originally Posted by JoeyCC View Post
    Hey guys,

    Firstly, Happy Thanks Giving for all of you who are Canadian! I've got a few problems with determinants which I can't solve! I'm hoping any of you can help me:





    I know the answer to the last problem of the first question is -23. Can anyone show me the rest? I thank you in advanced!
    for (i):

    \det(\text{adj}(A^T))=\det((\text{adj}(A))^T)=\det  (\text{adj}(A))=(\det (A))^2=25. in general for an n \times n matrix A we have \det(\text{adj}(A))=(\det(A))^{n-1}.

    \det(\text{adj}(A^{-1}))=\frac{1}{(\det(A))^2}=\frac{1}{25}.

    \det(\text{adj}(8A))=8^6(\det(A))^2=8^6 \times 25=6553600.


    for the first part of (ii) see my answer to the first part of (i). for the second part do a couple of elementary row opertaions to show that:

    \det \begin{bmatrix} v_1 \\ 7v_2 + 2v_4 \\ v_3 \\ 4v_2 + 5v_4 \end{bmatrix} = \det \begin{bmatrix} v_1 \\ 7v_2 \\ v_3 \\ \frac{27}{7}v_4 \end{bmatrix}=27 \det \begin{bmatrix} v_1 \\ v_2 \\ v_3 \\ v_4 \end{bmatrix}= 27 \times 6 = 162.
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