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. and

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: If B had an inverse given by. We have ; that is, which implies that 0=1-an impossibility! Thus B does not have an inverse

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: A does not have an inverse. we draw the conclusion that A is singular.

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: So (–2, 3) is the solution. Multiply by the inverse

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.

11. Solution: Let x, y, and z (in millions of dollars) be the amount awarded to organization I, II, and III, respectively. Then we have Put 0.6x + 0.4y + 0.2z = 9.2 (8.2) 0.3x + 0.3y + 0.6z = 9.6 (7.2) 0.1x + 0.3y + 0.2z = 5.2 (3.6). The quantities within the brackets are for part (b). We can rewrite the systems as AX = B 1, and AX =

12. . a. that is, x = 6, y = 10, and z = 8, and Organization I will receive $6 million, Organization II will receive $10 million, and Organization III will receive $8 million. b. that is, x = 8, y = 6, and z = 5, and Organization I will receive $8 million, Organization II will receive $6 million, and Organization III will receive $5 million.