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Math Help - Matrices - please help!

  1. #1
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    Matrices - please help!

    Here are two sets of numbers. You must calculate the answer and decipher the code. You do not need a calculator, although it does make it much easier.

    Here are the sets of numbers.
    [ 101 -42 ]
    [ 453 255]
    [ 345 92 ]
    [801 471 ]



    [ 6 -5 ] to the -1
    [ 29 18 ]

    the first is the code, and you have to multiply it by the inverse of the second.

    Inverse=1/determinant x matrix

    to find the determinant, do the topleft corner times the bottomright corner, minus the other two corners multiplied.
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  2. #2
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    To find the inverse of the second matrix consider

     <br />
\left[ \begin{array}{cc}<br />
a & b \\<br />
c & d \end{array} \right]^{-1} = \frac{1}{ad-bc}\left[ \begin{array}{cc}<br />
d & -b \\<br />
-c & a \end{array} \right] <br />
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  3. #3
    MHF Contributor

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    Quote Originally Posted by everydaysahollyday View Post
    Here are two sets of numbers. You must calculate the answer and decipher the code. You do not need a calculator, although it does make it much easier.

    Here are the sets of numbers.
    [ 101 -42 ]
    [ 453 255]
    [ 345 92 ]
    [801 471 ]



    [ 6 -5 ] to the -1
    [ 29 18 ]

    the first is the code, and you have to multiply it by the inverse of the second.
    Is the question given in exactly that way? If, in fact, "the second" is \left[\begin{array}{cc}6 & -5 \\ 29 & 18\end{array}\right]^{-1} then "the inverse of the second" is just \left[\begin{array}{cc} 6 & -5 \\ 29 & 18\end{array}\right] itself! The "inverse of the inverse" is just the matix itself.

    Inverse=1/determinant x matrix

    To find the determinant, do the topleft corner times the bottomright corner, minus the other two corners multiplied.
    Last edited by mr fantastic; January 15th 2010 at 01:32 AM. Reason: Fixed latex
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  4. #4
    Newbie bletchley's Avatar
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    It appears to be a simple cryptography application.

    A secret message (or plaintext) was placed into an unknown matrix A using either ASCII or simple substitution (1=A, 2=B ... 26=Z). It was then multiplied by the second matrix, the encoding matrix, to produce the first matrix you gave us (the ciphertext). Your instructor has was kind enough to give you the encoding matrix, the second one you see. So she wants you to find the decoding matrix, which is done by simply finding the inverse of the encoding matrix. After that, simply multiply the decoding matrix by the first matrix (the ciphertext) to produce the original plaintext matrix A.


    Finding the Inverse of a Matrix:
    <br />
\left[ \begin{array}{cc}a & b \\c & d \end{array} \right]^{-1} = \frac{1}{ad-bc}\left[ \begin{array}{cc}d & -b \\-c & a \end{array} \right]<br />

    Encryption:
    <br />
C=M*E<br />

    Decryption:
    <br />
D=E^{-1}<br />

    <br />
M=C*D<br />

    You were given C and E. Find D and then M.

    <br />
\left[ \begin{array}{cc}6 & {-5} \\29 & 18 \end{array} \right]^{-1} = \left[ \begin{array}{cc}\frac{18}{253} & \frac{5}{253} \\{-}\frac{29}{253} & \frac{6}{253} \end{array} \right]<br />

    <br />
\left[ \begin{array}{cc}101 & {-42} \\453 & 255 \\345 & 92\\801 & 471\end{array} \right] * \left[ \begin{array}{cc}\frac{18}{253} & \frac{5}{253} \\{-}\frac{29}{253} & \frac{6}{253} \end{array} \right]  = \left[ \begin{array}{cc}12 & 1 \\3 & 15 \\14 & 9\\3 & 27\end{array} \right]<br />
    The secret message is laconic.

    la·con·ic (lə-kŏn'ĭk) adj. Using or marked by the use of few words; terse or concise.
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