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Math Help - Square matrices within a matrix

  1. #1
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    Square matrices within a matrix

    Prove that if M is a n x n matrix that can be written in  M = \left({\begin{array}{cc} A & B \\ 0 & C \end{array}} \right)

    where A and C are square matrices, then det(M) = det(A) det(C)
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  2. #2
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    Write the matrix A in Jordan normal form with \lambda_1, ..., \lambda_k on the diagonal and 0's below the diagonal. Then  \text{det}(A) = \lambda_1 \lambda_2 ...\lambda_k. Likewise matrix C can be written in this form and we know \text{det}(C) = \lambda_{k+1}\lambda_{k+2}...\lambda_n .

    But then the entire matrix M is in Jordan normal form with \lambda_1, ..., \lambda_n on the diagonal and 0's below the diagonal, thus \text{det}(M) = \lambda_{1}\lambda_{2}...\lambda_n = \text{det}(A)\text{det}(C).
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