University:
California State University, Northridge
Course:
MATH 103 | Mathematical Methods for Business
Academic year:

2007
Views:

488

Pages:

13
Author:

problede1ad

4.5 Inverse of a Square Matrix In this section, we will learn how to find an inverse of a square matrix (if it exists) and learn the definition of the identity matrix. a Identity Matrix for Multiplication: The number 1 is called the multiplicative identity for real numbers: a(1) = a For example 5(1)=5  • In the set of nxn matrices we have a multiplicative inverse also: AI=A=IA • 2x2 case ! 1 0 $ ! a11 a12 $ = #" 0 1 &% #" a & a22 % 21 ! a11 a12 $ ! 1 0 $ = #" a & # & a22 % " 0 1 % 21 Identity matrices  3 x 3 identity matrix ! a11 a12 a13 $ ' 1 0 0 * , # &) # a21 a22 a23 & i )0 1 0 , = #" a a a &% )0 0 1 , + 31 32 33 ( ' 1 0 0 * ! a11 a12 a13 $ ) ,# & )0 1 0 , # a21 a22 a23 & = )0 0 1 , #" a a a &% ( + 31 32 33 !1 0 0 " #0 1 0 $ # $ #%0 0 1 $& Inverse of a real number  All real numbers (excluding 0) have an inverse.  For example 1 a! =1 a 1 5! = 1 5 What about matrices?  Some (not all) square matrices also have matrix inverses . ! 1 3$ ! '2 3 $ #" 1 2 &% #" 1 '1&% = ! 0 0 $ ! a11 a12 $ = #" 0 1 &% #" a & a22 % 21 Matrix Inverses    Some (not all) square matrices also have matrix inverses !1 If the inverse of a matrix A, exists, we call it A !1 !1 Then, A" A = A " A = I n Inverse of a 2 x 2 matrix  There is a simple procedure to find the inverse of a two by two matrix. This procedure only works for the 2 x 2 case. An example will be used to illustrate the procedure: A simple procedure to find the inverse of a 2 x 2 matrix.   This procedure only works for the 2 x 2 case. Find the inverse of ! a11 A=# " a21     a12 $ a22 &% Det(A) =difference of product of diagonal elements= a11a22 ! a21a12 In order for the inverse of a 2 x 2 matrix to exist, Det(A) cannot equal to zero. If Det(A)=0, then we conclude the inverse does not exist and we stop all calculations. If Det(A) is non-zero then we proceed: 1 " a22 A = Det(A) $# !a21 !1 !a12 % a11 '& Inverse of a two by two matrix !2 3" #1 2 $ % & General procedure to find the inverse matrix  We use a more general procedure to find the inverse of a 3 x 3 matrix. Problem: Find the inverse of the matrix " 1 !1 3 # $ 2 1 2% $ % $& !2 !2 1 %' Steps to find the inverse of any matrix 1. Augment this matrix with the 3 x 3 identity matrix. 2. Use elementary row operations to transform the matrix on the left side of the vertical line to the 3 x 3 identity matrix. The row operation is used for the entire row so that the matrix on the right hand side of the vertical line will also change. 3. When the matrix on the left is transformed to the 3 x 3 identity matrix, the matrix on the right of the vertical line is the inverse. Procedure Final result The inverse matrix  The inverse matrix appears on the right hand side of the vertical line and is displayed below.