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Math Help - Show that matrix A is nonsingular.

  1. #1
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    Exclamation Show that matrix A is nonsingular.

    I understand the concepts of singularity, but i got stuck in this problem.

    "Let A=[a,b,c,d] (a is 11, b is 12, c is 21, and d is 22 in their positions)

    ...Show that A is nonsingular if and only if ad-bc is NOT equal to zero"

    I multiplied A by a 2x2 matrix that has x1, x2, and y1 and y2 as its components and set it equal to an identity matrix but the system of equations doesnt lead me to anything resembling ad-bc not equal to zero.

    Thanks in advance.
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  2. #2
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    If the matrix A = \left[\begin{matrix} a & b\\ c & d\end{matrix}\right] is nonsingular, then it has an inverse.

    So if you go through the process of setting up the augmented matrix and performing row operations to find the inverse, the inverse is

    A^{-1} = \left[\begin{matrix}\phantom{-}\frac{d}{ad - bc} & -\frac{b}{ad - bc}\\ -\frac{c}{ad - bc} & \phantom{-}\frac{a}{ad - bc}\end{matrix}\right] = \frac{1}{ad - bc}\left[\begin{matrix} \phantom{-}d & -b \\ -c & \phantom{-}a\end{matrix}\right].

    Since you can never divide by 0, that means that the matrix will have an inverse as long as the stuff in the denominator, ad - bc \neq 0.
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