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Math Help - inverses of matrices

  1. #1
    Senior Member DivideBy0's Avatar
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    inverses of matrices

    I know that:
    \begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1}=\frac{1}{Determinant} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}

    But what is the equivalent expression for 3x3 (or higher dimensional matrices)?

    Thanks
    Last edited by DivideBy0; October 26th 2007 at 10:48 AM.
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    Quote Originally Posted by DivideBy0 View Post
    I know that:
    \begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1}=\frac{1}{Determinant} \begin{bmatrix} d & -b \\ -c & d \end{bmatrix}

    But what is the equivalent expression for 3x3 (or higher dimensional matrices)?

    Thanks
    The best way to find inverses is by Gaussian-Jordan elimination. Because that is the most efficient algorithm. However, it is not pleasant. There is a very elegant way to find inverses but when matrices get large it because computationally difficult to do (still it has important theoretical uses).

    Step 1: Given A a n\times n matrix compute \det A. If this determinant is zero then it have no inverse.

    Step 2: Compute the cofactor matrix. Meaning for each a_{ij} entry in the matrix compute the cofactor of a_{ij}. Replace each entry by its cofactor. This will form a matrix called the cofactor matrix.

    Step 3: Compute the adjoint matrix. To do this find the transpose of the cofactor matrix. The transpose operation on M (denoted by M^T) is flipping the elements along the main diagnol. So for the a_{ij} entry replace by a_{ji} entry. (Note, that the main diagnol of the matrix is unchanged by transpose operation).

    Step 4: Now divide each element in the adjoint matrix by the determinant. The resulting matrix is the inverse.
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  3. #3
    Senior Member DivideBy0's Avatar
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    Wow thanks... that is a lot to do to simply find an inverse...

    Maybe that's why my book didn't explain how to do it.

    Does anyone know how to find the inverse on a calculator?
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    Quote Originally Posted by DivideBy0 View Post
    Wow thanks... that is a lot to do to simply find an inverse...

    Maybe that's why my book didn't explain how to do it.

    Does anyone know how to find the inverse on a calculator?
    I do not own any of those new fancy show-off calculators. But I remeber from playing around with one of the TI's that I found a way to find inverse of matrices. There is a matrix command bottom. Program your matrix then press in the matrix into the screen followed by \boxed{^{-1}} and then enter. This will give you a matrix which is the inverse.
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