Inverse of a 3 x 3 matrix
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Inverse of a 3 times 3 matrix
There are two ways of finding the inverse of a 3 ´ 3 non-singular, square matrix. One technique uses Gaussian row reduction, the other a formula involving determinants. Technique of Gaussian row reduction Let A be a non-singular matrix, and A-1 its inverse This technique builds on the fact that [Equation goes here - download the original to see it.] Example [Equation goes here - download the original to see it.] Answer [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Swapping the rows around [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] In order to perform this technique you have to keep a level head whilst adding and multiplying the rows! Unfortunately, whilst this technique will now work with every problem involving matrices with numerical entries, it may not be easy to use when a more abstract problem is set. Formula for an inverse of a non-singular matrix To employ this formula you must understand the symbol for the determinant. This is [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it. Example [Equation goes here - download the original to see it. Then [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] This formula appears rather unwieldy. However, there is a way of making it easier to remember. This will also be better grasped if we first outline the proof that is indeed the inverse of A. To show this we must demonstrate that [Equation goes here - download the original to see it.] To show [Equation goes here - download the original to see it.]we have to multiply the matrices on the left and evaluate each term. We shall just indicate how this is done. Let us mark the entries of as follows, and consider each entry separatel [Equation goes here - download the original to see it.] That is, [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Similarly we could show that all the other non-diagonal entries evaluate to 0. Hence [Equation goes here - download the original to see it.] To recall the formula we firstly need to note two properties of matrices and determinants.1. if any two rows or columns of a non-singular matrix A are interchanged to give a matrix then det[Equation goes here - download the original to see it.]. Interchanging the rows or columns of a matrix has the effect of multiplying the determinant by -1.2. Cyclically permuting the rows or columns of a matrix A does not alter the determinant of A. To illustrate the first property, take our matrix [Equation goes here - download the original to see it.] To illustrate the cyclical permutation property let us permute the rows of A as follows. [diagram] [Equation goes here - download the original to see it.] to give [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Then, permuting the rows of A we obtain [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.][Equation goes here - download the original to see it.]
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Inverse of a 3 x 3 matrix
1 Inverse of a 3 times 3 matrix
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