In the basic mathematical operations of addition/ subtraction, multiplication/division we have identity elements and inverses. In addition and subtraction, the identity element is 0. If you add 0 to any number, you get the same number back again. x + identity = x (0 is the identity element in addition) The identity element in multiplication is 1. If you multiply anything by 1, you get your original element. 5 x identity = 5 (1 is the identity element in multiplication) The same thing is true with matrices. Any square matrix (same number of rows and columns) will have an identity matrix where you have 1's along the diagonal from top left to bottom right and zeros everywhere else. If you multiply a matrix by its identity matrix, you get the original matrix back. See the notes and videos for specific examples. Just like basic mathematical operations, matrices also have inverse matrices. In basic addition, the inverse element is the number you add to get the identity element of 0. Ex: 5 + inverse = 0(identity). So the inverse of 5 would be -5. -2 + inverse = 0(identity). So the inverse of -2 is +2. In multiplication, the inverse element for a number is its reciprocal. 5 x inverse = 1(identity). So the inverse of 5 for multiplication is 1/5. -1/2 x inverse = 1(identity). So the inverse for -1/2 is -2. -(1/2) x(-2) = 1(identity element). Matrices, if they are square matrices (same number of rows and columns), also have inverse matrices. If you multiply a matrix by its inverse, the result is the identity matrix. In math terms, Given a matrix A and its inverse matrix A^-1 then A x A^-1 = I (where I is the identity matrix) We can compute the inverse matrix of a given matrix. There are a few ways to do this. For a 2 x 2 matrix, there is a simple formula that can be used. For any square matrix whether 2 x 2 or larger, you can find the inverse by using Gauss-Jordan elimination on a matrix while simultaneously applying the same steps to the identity matrix. The assignments for this next section will involve finding inverse matrices of a given matrix. Homework due Monday, 5/18/20 pages 697-98: Numbers 4, 6, 8, 10, 12, 16 Videos Finding and identifying identity matrices www.youtube.com/watch?v=hPAS6H6xFa0 Formula for finding the inverse of a 2 x 2 matrix www.youtube.com/watch?v=oXV66LG5xUA Finding inverses using the calculator www.youtube.com/watch?v=_fFj4NbLcTU Finding inverse matrices using an augmented matrix (Gauss - Jordan) for sizes larger than 2 x 2 matrices www.youtube.com/watch?v=KBYvP6YG58g Example finding the inverse of a 3 x 3 matrix using Gauss - Jordan www.youtube.com/watch?v=NXCz9dw5Ce0&feature=youtu.be www.youtube.com/watch?v=6KEmzBwCBzI&feature=youtu.be
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