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2-6 - Chapter 2.6 Matrix Inverses In the last section we...

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Chapter 2.6: Matrix Inverses In the last section, we saw that a system of equations could be written as the matrix equation AX = B , where A a matrix of coefficients, B is a column matrix of constants, and X is a column matrix of variables. If we were able to solve this matrix equation for X , we would find the solution to the given system. We will look at how we can accomplish this today. To get an idea of what we are going to do, let’s first look at solving an equation with one variable. Example: Suppose we wish to solve the equation 5 x = 20. One way to solve this equation would be to divide both sides of the equation by 5. This would give us: 5 x 5 = 20 5 x = 4 We could rewrite our method of solution in terms of multiplication. In- stead of dividing both sides of the equation by 5, we could instead multiply both sides of the equation by 1 5 : parenleftbigg 1 5 parenrightbigg (5 x ) = parenleftbigg 1 5 parenrightbigg (20) parenleftbigg 1 5 parenrightbigg (5) ( x ) = 4 (1) ( x ) = 4 x = 4 The product of 1 5 and 5 is 1. When two numbers have a product of 1, we say that they are multiplicative inverses of each other. We will make a similar definition for matrices. Recall that the identity matrix I behaves like the number 1. We will say that two matrices are inverses of each other if their product is I . For numbers, every number other than zero has a multiplicative in- verse. The same is not true for matrices. 1
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Only some square matrices have inverses. Matrices which are not square do not have an inverse. If a square matrix A has an inverse, we call the inverse A - 1 . A square matrix that has an inverse is said to be nonsingular , and a square matrix that does not have an inverse is said to be singular .
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