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Math Help - Question about determinants

  1. #1
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    Question about determinants

    Question 1: Let A be a k x k matrix and let B be an (n-k) x (n-k) matrix. Let

    E= \begin{array}{cc} I_k & 0 \\ 0 & B\end{array}

    Show that det(E) = det(B)

    My attempt: det(E)=det(BI_k) - det(00)=det(B)det(I_k)=det(B)*1=det(B)

    I'm not sure this is correct, as I don't know the rules for finding the determinant of a matrix populated itself with matrices. Can anyone point me in the right direction?

    Thanks,
    CP
    Last edited by CrispyPlanet; November 28th 2013 at 01:03 PM.
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  2. #2
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    Re: Question about determinants

    Quote Originally Posted by CrispyPlanet View Post
    Question 1: Let A be a k x k matrix and let B be an (n-k) x (n-k) matrix. Let

    E= \begin{array}{cc} I_k & 0 \\ 0 & B\end{array}

    Show that det(E) = det(B)

    My attempt: det(E)=det(BI_k) - det(00)=det(B)det(I_k)=det(B)*1=det(B)

    I'm not sure this is correct, as I don't know the rules for finding the determinant of a matrix populated itself with matrices. Can anyone point me in the right direction?

    Thanks,
    CP
    One way you could show this is realizing that B is diagonizable by similarity transforms. Those (n-k)x(n-k) transform matrices can be extended to k x k by filling off diagonal elements with 0 and diagonal elements with 1 and using the new k x k transform matrix on E won't affect the Ik block. Then what you will have is a the product of a unitary matrix transpose with a diagonal matrix, the elements being the eigenvalues of E, and a unitary matrix. The determinant of this is the product of the matrix determinants which is 1 * det(E) * 1.

    det(E) is read off as the product of it's diagonal elements which is just 1n * det(B) = det(B)

    There are probably easier ways. See this and look for the section on Block matrices
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  3. #3
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    Re: Question about determinants

    Hi romsek, thanks for your response. Unfortunately I have just started linear algebra and a lot of that went over my head. I'll take a look at your link and try some other approach.

    Best,
    CP
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  4. #4
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    Re: Question about determinants

    try induction on k then
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