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Math Help - Linear question

  1. #1
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    Linear question

    I am looking at this proof:
    Let A be an n x n matrix and c a scalar. Show that det(cA)=c^ndet A.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by chadlyter View Post
    I am looking at this proof:
    Let A be an n x n matrix and c a scalar. Show that det(cA)=c^ndet A.
    Suppose it true for n=k, now consider any (k+1)x(k+1) matrix A, and expand
    about the first row:

    det(cA) = sum_{r=1 to k+1} (-1)^(r-1) c A_{1,r} det (cB)_{1,r}

    where (cB)_{1,r} is the matric obtained from cA by deleting the 1st row and
    the r-th col. Then by assumption:

    det(cA) = sum_{r=1 to k+1} (-1)^(r-1) c^{k+1} A_{1,r} det (B)_{1,r}

    ...........= c^{k+1} det(A)

    So if its true for n=k its true for n=k+1. It is trivialy true for n=1.

    Hence we conclude by mathematical induction that it is true for all
    positive integers.

    RonL
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  3. #3
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    Quote Originally Posted by chadlyter View Post
    I am looking at this proof:
    Let A be an n x n matrix and c a scalar. Show that det(cA)=c^ndet A.
    Here is a theoerem that you can factor a multiple out of each row or colomun.

    Say you want to
    [ 2 4]
    [3 5]
    You can factor a 2 to get,
    [1 2]
    [3 5]
    Then then find determinant of that and multiply by the number you factored out.

    But, the matrix c*A is a matrix were all entries are multipled by c.
    Thus, you factor c from each row, there are n rows, thus it is:
    c*c*...*c --> n times.
    Meaning,
    c^n
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