Inverse Matrices: Definition and Properties

Inverse Matrix Inverse matrix is a mathematical tool used in linear algebra, which finds the inverse of agiven matrix. The inverse matrix of a square matrix A is defined to be another squarematrix B such that AB = BA = I, where I denotes the identity matrix. The inverse of matrix A, by definition, is a matrix M whose property is that, when multipliedby A, it yields the identity matrix. If I multiply M by A and get identity, then it follows that the two properties are, in fact,equivalent. So if you apply for transformation A first, then M, you actually undo the transformation A,and vice versa. These two transformations are opposites of each other, inverse of eachother. So, to make sense of this, we need A to be a square matrix—that is, it must have thesame number of rows as columns (size n by n). Linear algebra is a general fact. You will see more in detail if you study linear algebra. Fornow you should just admit that. We denote the matrix M by A inverse, like this. OK? And then what is it good for? Well, forexample, the solution to a linear system--so what's a linear system in our newlanguage--it's a matrix times some unknown vector X equals some known vector B. Sohow do we solve that? We can solve for X by computing A inverse B. Why does that work? From there, we askhow we can get from one point on our graph to another. Let's be cautious. OK, well, I'm going to re-use this matrix, but I'm going to erase it andrewrite it anyway. So, AX is equal to B. We multiplied both sides of the equation by the inverse of A. Ainverse times AX is A inverse B. After that, A inverse times A is identity, so we have that Xequals A inverse B. To solve a system of equations, first multiply both sides of each equation by its inverse andthen subtract the two equations to get rid of any constants. If you have a calculator that can invert matrices, then it can solve linear systems quickly.Now, we should still learn how to compute these things by hand. Notice, if we know that A inverse will be on the left of B and not after it, we can confirm thisby reproducing the calculation. It's exactly in this derivation. I mean, to get from here to here, I multiplied the elements on the left by the inverse. Theresult was that those were simplified. If I had put A inverse on the right, the equation wouldbe AXA inverse, which may not make sense, and even if it does make sense, it doesn'tsimplify. The basic rule is that if there's an A on both sides of the equation, you have to multiply byA inverse on one side so that it cancels with the A on the other side.

So, just remember that if you have a square matrix times a column vector, the productmakes sense when the matrix is on the left and the vector is on the right. The other one does not work. It is not sized correctly. If A is a square matrix and X is acolumn vector, then this product makes sense. The other one does not make sensebecause it is not sized correctly. Nice.