Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Transcript

Inverse Matrices (2 x 2) How to find the inverse of a 2x2 matrix Inverse of a number When we are talking about our natural numbers, the inverse of a number is it’s reciprocal. When we multiply a number by it’s inverse we get 1. For example: 3 1 1 3 4 0.25 1 k k 1 1 Inverse of a matrix What do you think we would get if we multiplied a matrix by it’s inverse? Try it on your calculator. 1 A A I A matrix multiplied by its inverse always gives us an identity matrix. Inverse of a matrix Not all matrices have an inverse. If the determinant of a matrix is 0, then it has no inverse and is said to be SINGULAR. All others are said to be NON-SINGULAR Inverse of a matrix Which of these have an inverse? 3 2 3 2 1 2 3 4 Finding Inverses 2x2 8 10 A 3 4 1 A A I a b Let A-1 = c d So AA-1 =I 8 10 a b 1 0 3 4 c d 0 1 Multiplying out gives.. 8a 10c 8b 10d 1 0 3 a 4 c 3 b 4 d 0 1 8a 10c 1 3a 4c 0 8b 10d 0 3b 4d 1 Can you solve these to work out A-1? 2 5 A 1.5 4 1 Finding Inverses 2x2 There is an alternative method. a b A c d 1 d b A ad bc c a 1 In words: ad-bc represents det(A). What would happen if this was zero? 1 det( A) d a -b b -c c d a •Take the original matrix. •Switch a and d. •Change the signs of b and c. •Multiply the new matrix by 1 over the determinant of the original matrix. Finding Inverses 2x2 Example: Find the inverse of A. A A A 1 1 4 2 4 10 10 4 1 2 (2)(10) ( 4)(4) 4 1 10 4 2 = 4 4 5 1 2 1 1 2 Finding Inverses 2x2