4-6 3x3 Matrices, Determinants, & Inverses

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Presentation transcript:

4-6 3x3 Matrices, Determinants, & Inverses

Objectives Evaluating Determinants of 3 x 3 Matrices Using Inverse 3 x 3 Matrices

Using a Graphing Calculator to find the Determinant Enter matrix T into your graphing calculator. Use the matrix submenus to evaluate the determinant of the matrix. 4 2 3 –2 –1 5 1 3 6 T = The determinant of the matrix is –65.

Verifying Inverses Determine whether the matrices are multiplicative inverses. 0.5 0 0 0 0 0.5 0 1 1 2 0 0 0 2 1 0 2 0 a. C = , D = 0.5 0 0 0 0 0.5 0 1 1 2 0 0 0 2 1 0 2 0 1 0 0 0 1 0 0 4 1 = Since CD I, C and D are not multiplicative inverses. = /

Continued (continued) b. A = , B = 0 0 1 0 1 0 1 0 –1 1 0 1 0 1 0 0 0 1 0 1 0 1 0 –1 1 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 0 1 = 0 0 1 0 1 0 1 0 –1 1 0 1 Since AB = I, A and B are multiplicative inverses.

Solving a Matrix Equation 2 0 1 0 1 4 1 0 0 –1 8 –2 Solve the equation. X = Let A = . 2 0 1 0 1 4 1 0 0 Find A–1. X = 0 0 1 –4 1 8 1 0 –2 –1 8 –2 Use the equation X = A–1C. Multiply. X = –2 –4 3

Real World Example Use the alphabet table and the encoding matrix. 0.5 0.25 0.25 0.25 –0.5 0.5 0.5 1 –1 matrix K = . a. Find the decoding matrix K–1. K–1 = Use a graphing calculator. 0 2 1 2 –2.5 –0.75 2 –1.5 –1.25

Continued (continued) 11.25 16.75 24.5 5.75 17 5.5 1.5 –12 15 11.25 16.75 24.5 5.75 17 5.5 1.5 –12 15 b. Decode . Zero indicates a space holder. = 0 2 1 2 –2.5 –0.75 2 –1.5 –1.25 11.25 16.75 24.5 5.75 17 5.5 1.5 –12 15 13 22 26 7 0 24 12 23 22 Use the decoding matrix from part (a). Multiply. The numbers 13 22 26 7 0 24 12 23 22 correspond to the letters NEAT CODE.

Homework Pg 209 # 5, 6, 8, 9, 10 ,11