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Math Help - show that the matrix is non singular if and only if this condition is met

  1. #1
    Member Jskid's Avatar
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    show that the matrix is non singular if and only if this condition is met

    Show that the matrix A=\left[\begin{matrix}a&b\\c&d\end{matrix}\right] is nonsingular if and only if ad-bc≠0. If this condition holds, show that \left[\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right]

    I think what I want to do is show that \left[\begin{matrix}a&b\\c&d\end{matrix}\right] is row equivalent to I_2. When I was doing this I got stuck at \left[\begin{matrix}ca&da\\0&1\end{matrix}\right]
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  2. #2
    Senior Member Tinyboss's Avatar
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    Showing that the inverse is given by that particular matrix (with ad-bc in the denominators) gives you one direction. To do the other, suppose ad-bc=0 and try multiplying the top row by d and the bottom row by b.
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  3. #3
    MHF Contributor

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    Show that \left[\begin{matrix}a&b\\c&d\end{matrix}\right]\left[\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right]= \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}.

    Since the inverse of a matrix is unique, it follows that the denominator of that fraction cannot be 0.
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