The Inverse of a matrix
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 2(2 matrix, calculation of the inverse of a matrix is straightforward and proceeds thusly: if
[pic]
then its inverse is given by
[pic]
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:
[pic]
Of course, it’s always a good idea to check one’s calculations, in this case by verifying that [pic]:
[pic] QED
But, there’s a catch, which you can appreciate if you look closely at the formula for the inverse, specifically at the expression for the value of[pic]. Because of that [pic], term, not all matrices can be inverted: if the determinant of a matrix is equal to zero, the matrix is termed singular, and can’t be inverted (because [pic]). For example, if we want to calculate the inverse of
[pic].
we first calculate the determinant and find that
[pic],
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
[pic].
First, try calculating the inverse for yourself. You should come up with
[pic]
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 2(2 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 3(3:
That should convince you to leave matrix inverse calculations to Matlab!
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