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Unformatted text preview: Atsushi Inoue ECG 765: Mathematical Methods for Economics Fall 2009 Lecture Notes 12 Matrix Inverse, Matrix Rank and the Fundamental Theorem of Linear Algebra When we solve constraint optimization problems, one of the conditions, called constraint qualification, is often written in terms of rank. The inverse of a matrix is often used in graduate courses especially in econometrics. We will review these important concepts in matrix algebra. Outline A. Matrix Inverse B. Matrix Rank C. Fundamental Theorem of Linear Algebra A. Matrix Inverse Definition Let A be a square n n matrix. We shall say that A is invertible or non-singular if there exists an n n matrix B such that AB = BA = I n . This matrix B is called the inverse of A and is denoted by A- 1 . The matrix A- 1 is called A inverse. The inverse is unique if it exists. Example. A = &quot; a b c d # . Theorem. (a) If A is invertible then so is A- 1 . The inverse of A- 1 is ( A- 1 )- 1 = A. 1 (b) If A and B are invertible then so is AB . The inverse of AB is ( AB )- 1 = B- 1 A- 1 . (c) If A is invertible, ( A- 1 ) &gt; = ( A &gt; )- 1 ....
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- Fall '07