CHAPTER 05.20: SYSTEM OF EQUATIONS: Using concept of inverse to solve a set of equations

In this segment we’ll talk about whether we can use the concept of inverse to solve a set of equations. Let’s suppose somebody gives you an equation of unknowns. So this is n rows n columns, n rows one column, n rows one column. And says that hey I’m going to give you the inverse of A so which means that he or she is telling you that the A inverse does exist. So the question arises, will it be able to solve for x, can you find the solution to the set of equations? So let’s see what happens. If I take A inverse which I’m assuming which is given to me that it exists. I can multiply both sides by A inverse. So I get A inverse here like this. But I know that the A inverse times A is nothing but the identity matrix. And this one is just A inverse times C. But then I know that if I multiply the identity matrix by a vector like this one, so this multiplication is legal, it’s allowed because the number of columns here is the same as the number of rows here. But I’ll end up with X itself because anytime a matrix is multiplied by the identity matrix and that matrix’s multiplication is allowed you end up with the matrix itself. So that’ll be equal to A inverse times n by n matrix times C matrix. So what were basically saying is that hey A inverse, if somebody gives you the inverse of the caution matrix you should be able to easily find out what the solution vector is because all you have to do is to write down the inverse matrix and put the C matrix right here. Multiply the two matrices and all you have to do is to do the multiplications and you end up with the solution of the set of equations. So that the good thing about understanding what it means to find the inverse of a matrix and how to apply it to solve a set of equations. And this is the end of this segment.