1. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. 2.6 Inverse of a Matrix Let A be a square matrix. A square matrix A -1 of equal size such that A -1 A = AA -1 = I is called the inverse of A. Example.

2. Not every square matrix has an inverse. A square matrix that has an inverse is said to be nonsigular. A matrix that does not have an inverse is said to be singular. Example: Reason:

3. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Finding the Inverse of a Matrix Given the square matrix A. 1.Adjoin the identity matrix I (of the same size) to form the augmented matrix: [A | I] 2.Use row operations to reduce the matrix to the form: [I | B] (if possible) Matrix B is the inverse of A. or use a graphing calculator.

4. Solution:

5. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Solution:

6. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Formula for the Inverse of a 2 x 2 Matrix Let Example.

7. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Solving a System of Equations Using an Inverse If AX = B is a linear system of equations (number of equations = number of variables) and A -1 exists, then X = A -1 B is the unique solution of the system.

8. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Example. Use an inverse matrix to solve:

9. Example: The Carver Foundation funds three nonprofit organizations Engaged in alternate-energy research activities. From past data, the Proportion of funds spent by each organization in research on solar energy, energy from harnessing the wind, and energy from the motion of Ocean tides is given in the accompanying table. Proportion of Money Spent Solar Wind Tides Organization I 0.6 0.3 0.1 Organization II 0.4 0.3 0.3 Organization III 0.2 0.6 0.2

10. Find the amount awarded to each organization if the total amount spent By all three organizations on solar, wind, and tidal research is a.$9.2 million, $9.6 million, and $5.2 million, respectively. b.$8.2 million, $7.2 million, and $3.6 million, respectively.