2. Ch. 4.5 2 X 2 Matrices, Determinants, and Inverses

3. Square Matrix A matrix with the same number of columns as rows

4. Multiplicative Identity Matrix For an n X n square matrix, the multiplicative identity matrix is an n X n square matrix I, or, with 1’s along the main diagonal and 0’s elsewhere.

5. Multiplicative Inverse of a Matrix If A and X are n x n matrices, and AX = XA = I, then X is the multiplicative inverse of A, written:

6. Show that matrices A and B are multiplicative inverses. A = B = 3 –1 7 1 0.1 0.1 –0.7 0.3 = (3)(0.1) + (–1)(–0.7) (3)(0.1) + (–1)(0.3) (7)(0.1) + (1)(–0.7) (7)(0.1) + (1)(0.3) AB = I, so B is the multiplicative inverse of A.

7. Check Understanding P. 196 # 1B

8. Determinate of a 2 x 2 matrix The Determinate of a 2 x 2 matrix is ad – bc.

9. Evaluate each determinant. a. det b. det c. det 7 8 –5 –9 4 –3 5 6 a –b b a = = (7)(–9) – (8)(–5) = –23 7 8 –5 –9 = = (4)(6) – (–3)(5) = 39 4 –3 5 6 = = (a)(a) – (–b)(b) = a 2 + b 2 a –b b a

10. Check Understanding P. 196 2 B and C

11. Inverse of a 2x2 Matrix Let A =. If det A  0, then has an inverse. If det A  0, then

12. Determine whether each matrix has an inverse. If it does, find it. Find det X. ad – bc = (12)(3) – (4)(9) Simplify. = 0 12 4 9 3 Since det X = 0, the inverse of X does not exist. Find det Y. ad – bc = (6)(20) – (5)(25) Simplify. = –5 6 5 25 20 Since the determinant 0, the inverse of Y exists.=/ a. X = b. Y =

13. (continued) = – 20 –5 Substitute –5 for the –25 6 determinant. 1515 = Multiply. –4 1 5 –1.2 Y –1 = 20 –5 Change signs. –25 6 Switch positions. 20 –5 Use the determinant to –25 6 write the inverse. = 1 det Y 1 det Y

14. Check Understanding P. 197 # 3A

15. Solve X = for the matrix X. The matrix equation has the form AX = B. First find A –1. 9 25 4 11 3 –7 A –1 = 1 ad – bc d –b –c a Use the definition of inverse. = 1 (9)(11) – (25)(4) 11 –25 –4 9 Substitute. = –11 25 4 –9 Simplify. Use the equation X = A –1 B. X = –11 25 4 –9 Substitute. 3 –7

16. (continued) = = (–11)(3) + (25)(–7) (4)(3) + (–9)(–7) Multiply and simplify. –208 75 Check: X = 9 25 4 11 Use the original equation. 3 –7 9 25 4 11 Substitute. 3 –7 –208 75 Multiply and simplify. 3 –7 9(–208) + 25(75) 4(–208) + 11(75) = 3 –7 3 –7

17. Homework P. 199 # 2-24 even