# Matrix_4 - Matrix Inverse Objectives: At the end of this...

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Matrix Inverse © Arizona State University, Department of Mathematics and Statistics 1 of 4 Objectives: At the end of this lesson, you should be able to: 1. Understand why we want to find a matrix inverse. 2. Understand why we cannot always find a matrix inverse. 3. Apply Gaussian Elimination to find a matrix inverse Background A critical piece in the use of matrices is the inverse . Let’s look back at how we used the multiplicative inverse to solve simple algebraic equations. Suppose you wanted to solve . You would probably think “divide by 4". However, notice that we never 4 8 x = defined a division process for matrices. Thinking back to your beginning algebra, you recall that you were told to apply the multiplicative inverse (usually called the reciprocal in arithmetic). The fancy way to do this is to write the reciprocal as . Then we get . After all that 1 4 - 1 1 4 4 1 4 8 x x - - = = . We can solve matrix equations exactly the same way. 2 x = The Inverse Matrix Defined Suppose someone told you to write the solution for where A, X, and B are matrices. IF (big if) we AX B = knew that there is a matrix out there that is the inverse of A , the solution would be . 1 1 A AX IX A B - - = = So let’s begin with a definition: For the n×n matrix A , if an n×n matrix B exists so that , we call it n AB BA I = = the inverse of A . We use the symbol A -1 to represent the inverse of A . We will tell you that B is unique. For each matrix A with an inverse, there is only one A -1 . There is a ton of stuff going on in this definition. The only way we can guarantee both a left and right side

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## This note was uploaded on 04/08/2012 for the course MATH 340 taught by Professor Davidglenn during the Spring '12 term at Boston Architectural.

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Matrix_4 - Matrix Inverse Objectives: At the end of this...

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