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Transcript

The Inverse of a matrix Let’s calculate B-1, the inverse of matrix B, just to give you an idea of what’s involved and point out a few things. For a general 22 matrix, calculation of the inverse of a matrix is straightforward and proceeds thusly: if a b B c d then its inverse is given by B 1 1 d b 1 d b det B c a ad bc c a where “det B” represents the determinant of B, something you probably remember, perhaps vaguely, from high-school algebra. Recall also that the determinant of a matrix is a scalar. Let’s work a specific example: 1 2 B det B 1 4 3 2 2 3 4 1 1 4 2 2 B 1 2 3 1 3 2 1 2 Of course, it’s always a good idea to check one’s calculations, in this case by verifying that BB 1 I : 1 2 3 1 1 1 0 1 2 2 BB 1 3 I 1 3 4 2 2 6 6 3 2 0 1 QED But, there’s a catch, which you can appreciate if you look closely at the formula for the inverse, 1 specifically at the expression for the value of det B . Because of that , term, not all ad bc matrices can be inverted: if the determinant of a matrix is equal to zero, the matrix is termed 1 1 singular, and can’t be inverted (because ). For example, if we want to calculate det( B) 0 the inverse of 3 1 Z . 1.8 5.4 we first calculate the determinant and find that det Z 1 5.4 1.8 3 5.4 5.4 0 , meaning Z is singular and we can’t invert it. We therefore can never carry out matrix division with Z as a divisor. Something else: multiplying a matrix by its inverse, which is the equivalent of dividing a matrix by itself, yields the identity matrix. Try this with the matrix 1 4 A . 3 2 First, try calculating the inverse for yourself. You should come up with 0.2 0.4 A 1 0.3 0.1 Then, if you carry out the multiplication of A by A-1, you should obtain the corresponding identity matrix (try it). Finally, although calculating the inverse of a 22 matrix is easy, for anything larger, it’s not. Here’s a link to a site that will walk you through the process for a 33: http://www.easycalculation.com/matrix/inverse-matrix-tutorial.php That should convince you to leave matrix inverse calculations to MATLAB!