Abstract Algebra MATH 3330
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This 5 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 3330 at University of Houston taught by Mary Flagg in Fall. Since its upload, it has received 14 views. For similar materials see /class/208404/math-3330-university-of-houston in Mathmatics at University of Houston.
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Date Created: 09/19/15
Math 3330 Section 16 Ova CD W Matrices An m X matrix A over a set S is a rectangular array of elements in S arranged in m rows and n columns Shorthand notation Aaijmxn Inlongform ICLH Orva CM QM an 39 quot aVMJ Dimensionan Square matriximn Diagonal matrix 7 a squ atrix With the nonzero entries only on the main diagonal Two matrices are equal ifi 5W 5 Z a c orre spo ndl M7 e AI iamp 56 5 W Matrix Addition ABC aiijxn bij1mxn Ciijxn CijaijbijIVin 0 1L 5 1 93 6 U Properties of Matrix Addition It is a binary operation on the set of all m by n matrices for any m and n It is associative An identity element exists Q O 1 m AI Ea j Every matrix has an additive inverse A y M 1 y 1 4 r 0k f a m M It is commutative ch NW is commuim r39vz Matrix Multiplication aik Jan X bkj1nxp Cij mxp Cijziaikbkj Q 3 I H l o Z 3 K SUB 3 h SW moctr ice s CAL0 40 LEWx Properties of Matiix Multiplication 3 Not a binary operation unless mm A a 1 L Associative e I An identity element exists for the set MIl S if S has an identity for multiplication Q Not all n by n matrices have inverses T I G Multiplication is NOT commutative A A O P Distn39butive Propelty holds for Mrl Aw M T A8 AC One 7 Sided Inverses I O l O O t O7 O 0 O O O c O 7 1 Example Find two nonzero matIices A and B such that AB 0 I echWM El 1 2 LetA andB 3 1 4 0 2 1 Find AB ABAB A2 B2 A2 B2 AJrQM E 1 73 Kg7 Ag 62 in i J 25 g dew Mad 1413 BR H9 et u exercise 20a amp q r O O H O 0gt 8 b 3 BB
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