Lesson 12 – 4 Inverses of Matrices Pg. 624 #1–13 odd, 16, 21, 23, 31, 39, 41 Lesson 12 – 4 Inverses of Matrices Pre-calculus To show that two matrices are inverses. To find the multiplicative inverse of a matrix

Learning Objective To show that two matrices are inverses. To find the multiplicative inverse of a matrix

Inverses of Matrices Remember if 𝐴 and 𝐵 are inverses, 𝐴𝐵=𝐼 and 𝐵𝐴=𝐼 *Only square matrices can have multiplicative inverses* 1. Show that matrix B is the multiplicative inverse of matrix A. 𝐴= 4 3 3 2 and 𝐵= −2 3 3 −4 4 3 3 2 −2 3 3 −4 = 1 0 0 1 𝐴𝐵= −2 3 3 −4 4 3 3 2 = 1 0 0 1 𝐵𝐴=

Inverses of Matrices 𝑎 𝑏 𝑐 𝑑 To find the inverse of a square 2 x 2 matrix, we: Step 1: Find the determinant 𝑎 𝑏 𝑐 𝑑 =𝑎𝑑 −𝑏𝑐 Step 2: Make some changes in your matrix: Switch 𝑎 & 𝑑 spots 𝑑 −𝑏 −𝑐 𝑎 Change the sign of 𝑐 & 𝑏 Step 3: Multiply 1 𝑑𝑒𝑡 𝑑 −𝑏 −𝑐 𝑎

Inverses of Matrices 2. Find the multiplicative inverse. 𝐴= 9 4 7 3 𝐴= 9 4 7 3 1 𝑑𝑒𝑡 𝑑 −𝑏 −𝑐 𝑎 𝑑𝑒𝑡 =9(3) −4(7) =27−28 =−1 1 −1 3 −4 −7 9 = −3 4 7 −9 𝐴 −1 = −3 4 7 −9

Inverses of Matrices For 3 x 3 and higher, we can use a graphing calc! 3. Find the multiplicative inverse. 𝐴= 4 −2 3 3 −5 −2 2 4 −3 MATRIX: 2nd x-1  EDIT  [A] Enter data QUIT: 2nd Mode MATRIX: 2nd x-1  Choose [A] 𝐴 −1 = 23 148 3 74 19 148 5 148 −9 74 17 148 11 74 −5 37 −7 74 [𝐴] −1 = Yikes!! Change to fractions MATH  1: Frac *arrow over to see the rest*

We can use inverses to solve for an unknown matrix Inverses of Matrices *Be careful of the order* If 𝐴, 𝑋, 𝑎𝑛𝑑 𝐵 are matrices, and 𝐴𝑋=𝐵 To “get rid” of 𝐴, we multiply by 𝐴 −1 𝐴 −1 (𝐴𝑋)= 𝐴 −1 𝐵 𝑋= 𝐴 −1 𝐵

Inverses of Matrices 𝐴𝑋=𝐵 𝑋= 𝐴 −1 𝐵 4. Solve for X 2 −3 4 −5 𝑋= 6 11 2 −3 4 −5 𝑋= 6 11 2 −3 4 −5 −1 = −5 3 −4 2 1 −10−(−12) 𝐴𝑋=𝐵 = 1 2 −5 3 −4 2 = −2.5 1.5 −2 1 𝑋= 𝐴 −1 𝐵 = 3 2 −1 𝑋= 1 2 −5 3 −4 2 6 11 = 1 2 3 −2 = 3 2 −1 𝑋= −2.5 1.5 −2 1 6 11 = 1.5 −1

Inverses of Matrices A𝑋=𝐵 𝑋= 𝐴 −1 𝐵 5. Solve for X. (Use your calculator!) Inverses of Matrices 1 1 1 1 −4 3 1 6 2 𝑋= 16 42 14 ↑ enter for matrix A ↑ enter for matrix B A𝑋=𝐵 𝑋= 𝐴 −1 𝐵 𝑋= 𝐴 −1 𝐵 𝑋= 10 −2 8 𝑋= [𝐴] −1 [𝐵]

We can take a system of equations and turn it into a matrix equation to then solve! Systems 6. Set up the matrices to solve the system 6𝑥−𝑦=22 2𝑥+𝑦=2 A𝑋=𝐵 coefficients answers You continue to solve like the previous examples  6 −1 2 1 = 22 2 𝑋 𝑋= 𝐴 −1 𝐵 ↑ represents 𝑥 𝑦 = 1 6−(−2) 1 1 −2 6 22 2 𝑋= 𝐴 −1 𝐵 = 1 8 1 1 −2 6 22 2 = 3 −4 (3, −4)

12-6 Check up 6 −3 2 10 15 1. Solve for X. Show work. No calculator. 5 −3 4 −2 𝑋= 5 10 12-6 Check up 2. Solve for X. Use calculator!! 3 4 −6 8 −2 7 5 3 4 𝑋= −6 68 29 6 −3 2 10 15

Assignment Pg. 624 #1–13 odd, 16, 21, 23, 31, 39, 41