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Math Help - if det(aM + K) = det(bM + K) where a, b are scalars and M, K are matrices, is a = b?

  1. #1
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    if det(aM + K) = det(bM + K) where a, b are scalars and M, K are matrices, is a = b?

    hello,

    i have a problem. if i know that

    <br />
\textnormal{det}(a \mathbf{M} + \mathbf{K}) = 0<br />

    <br />
\textnormal{det}(b \mathbf{M} + \mathbf{K}) = 0<br />

    where a and b are scalars and \mathbf{M} and \mathbf{K} are matrices,
    does this mean that a = b?

    also, \mathbf{M} is diagonal, if that helps.

    cheers for reading and sorry if this is a dumb question,
    jack.
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  2. #2
    MHF Contributor Swlabr's Avatar
    Joined
    May 2009
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    1,176
    Quote Originally Posted by thepartymushroom View Post
    hello,

    i have a problem. if i know that

    <br />
\textnormal{det}(a \mathbf{M} + \mathbf{K}) = 0<br />

    <br />
\textnormal{det}(b \mathbf{M} + \mathbf{K}) = 0<br />

    where a and b are scalars and \mathbf{M} and \mathbf{K} are matrices,
    does this mean that a = b?

    also, \mathbf{M} is diagonal, if that helps.

    cheers for reading and sorry if this is a dumb question,
    jack.
    Look at the 2x2 case. Assuming a=1 then we need to solve the equation
    (b-1)(n(p+bm+m)+sm) = 0 where p is the top left coordinate of the matrix K, s the bottom left. m, n are the coordinates of M. Assuming b is not 1 then we want to find n, m, b, p, s such that n(p+bm+m)+sm=0. So, let s=0=n.

    That is,  M =<br />
\left( \begin{array}{ccc}<br />
m & 0 \\<br />
0 & 0 \end{array} \right) and  K =<br />
\left( \begin{array}{ccc}<br />
p & q \\<br />
r & 0 \end{array} \right)

    is a counter-example.
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  3. #3
    Newbie
    Joined
    Sep 2009
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    thanks, this helped get me thinking... what if the diagonal entries are positive? what if a and b are both negative? etc. seems like a bit of a long shot though.

    thanks again,
    jack.

    p.s. i assume you meant "s is the bottom right".
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