Finding Inverse Matrices

Finding Inverse Matrices - Finding Inverse Matrices In the...

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Finding Inverse Matrices In the previous section we introduced the idea of inverse matrices and elementary matrices. In this section we need to devise a method for actually finding the inverse of a matrix and as we’ll see this method will, in some way, involve elementary matrices, or at least the row operations that they represent. The first thing that we’ll need to do is take care of a couple of theorems. Theorem 1 If A is an matrix then the following statements are equivalent. (a) A is invertible. (b) The only solution to the system is the trivial solution. (c) A is row equivalent to . (d) A is expressible as a product of elementary matrices. Before we get into the proof let’s say a couple of words about just what this theorem tells us and how we go about proving something like this. First, when we have a set of statements and when we say that they are equivalent then what we’re really saying is that either they are all true or they are all false. In other words, if you know one of these statements is true about a matrix A then they are all true for that matrix. Likewise, if one of these statements is false for a matrix A then they are all false for that matrix. To prove a set of equivalent statements we need to prove a string of implications. This string has to be able to get from any one statement to any other through a finite number of steps. In this case we’ll prove the following chain . By doing this if we know one of them to be true/false then we can follow this chain to get to any of they others. The actual proof will involve four parts, one for each implication. To prove a given implication we’ll assume the statement on the left is true and show that this must in some way also force the statement on the right to also be true. So, let’s get going. Proof : : So we’ll assume that A is invertible and we need to show that this assumption also implies that will have only the trivial solution. That’s actually pretty easy to do. Since A is invertible we know
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that exists. So, start by assuming that is any solution to the system, plug this into the system and then multiply (on the left) both sides by to get, So, has only the trivial solution and we’ve managed to prove this implication. : Here we’re assuming that will have only the trivial solution and we’ll need to show that A is row equivalent to . Recall that two matrices are row equivalent if we can get from one to the other by applying a finite set of elementary row operations. Let’s start off by writing down the augmented matrix for this system. Now, if we were going to solve this we would use elementary row operations to reduce this to reduced row-echelon form, Now we know that the solution to this system must be, by assumption. Therefore, we also know what the reduced row-echelon form of the augmented matrix must be since that must give the above solution. The reduced-row echelon form of this augmented matrix must be,
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Finding Inverse Matrices - Finding Inverse Matrices In the...

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