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Math Help - Determinant of two matrices

  1. #1
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    Determinant of two matrices

    Hi,

    I didn't get so well what this problem is asking. Mostly how the B matrix is generated.

    Let \lambda \in R. The matrix B is generated by the matrix A by addition of the \lambda multiple of the j-th row to the i-th row ( i \neq j). Show that detB = detA.
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  2. #2
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    Quote Originally Posted by TheFangel View Post
    Hi,

    I didn't get so well what this problem is asking. Mostly how the B matrix is generated.

    Let \lambda \in R. The matrix B is generated by the matrix A by addition of the \lambda multiple of the j-th row to the i-th row ( i \neq j). Show that detB = detA.
    For example, if A= \begin{pmatrix}1 & 3 & 3 \\ 4 & 5 & 1 \\ 3 & 2 & 3\end{pmatrix} and B "is generated by the matrix A by addition of the \lambda multiple of the 3rd row the the first row" (taking j= 3 and i= 1), then B= \begin{pmatrix}1+ 3\lambda & 3+ 2\lambda & 3+ 3\lambda \\ 4 & 5 & 1\\ 3 & 2 & 3 \end{pmatrix}.

    A determinant is a sum of products, each of which involves exactly one factor from each row. Here, such a product, a_{1i}a_{2j}a_{3k}... will be replace by a_{1i}a_{2j}(a_{3k}+ \lambda a_{nk}).... Multiply that out.
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  3. #3
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    Wow!
    Thanks very much!
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