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Math Help - Get just a few values of invertible matrix

  1. #1
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    Get just a few values of invertible matrix

    \left(\begin{array}{ccccc}4&1&0&0&0\\\end{array}\r  ight) \left(\begin{array}{ccccc}s&-1&0&0&0\\0&s&-1&0&0\\0&0&s&-1&0\\0&0&0&s&-1\\0&0&4&4&s-1\\\end{array}\right)^{-1}  \left(\begin{array}{c}0\\0\\0\\0\\1\\\end{array}\r  ight)

    If I want to multiply the above 3 matrices, then I only need to find the top-right, and the entry below it in the inverse matrix (as all others will be zero).

    How do I go about finding only a few entries in a large inverted matrix like this?
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  2. #2
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    Call the matrix S, and let v = \begin{bmatrix}0\\0\\0\\0\\1\end{bmatrix}. If x = S^{-1}v then Sx = v. if x = \begin{bmatrix}a\\b\\c\\d\\e\end{bmatrix} then the equation Sx = v becomes \begin{bmatrix}sa-b\\ sb-c\\ sc-d\\ sd-e\\ 4c+4d+(s-1)e\end{bmatrix} = \begin{bmatrix}0\\0\\0\\0\\1\end{bmatrix}. That gives you five easily solved equations telling you that a = \frac1{s^5-s^4+4s^3+4s^2}.

    The expression \begin{bmatrix}4&1&0&0&0\end{bmatrix}x then becomes a(4+s) = \frac{s+4}{s^5-s^4+4s^3+4s^2}.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by bumcheekcity View Post
    \left(\begin{array}{ccccc}4&1&0&0&0\\\end{array}\r  ight) \left(\begin{array}{ccccc}s&-1&0&0&0\\0&s&-1&0&0\\0&0&s&-1&0\\0&0&0&s&-1\\0&0&4&4&s-1\\\end{array}\right)^{-1} \left(\begin{array}{c}0\\0\\0\\0\\1\\\end{array}\r  ight)

    If I want to multiply the above 3 matrices, then I only need to find the top-right, and the entry below it in the inverse matrix (as all others will be zero).

    How do I go about finding only a few entries in a large inverted matrix like this?
    google for block matrix inverse

    CB
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