Lect1-05-web

Lect1-05-web - MATH 304, Fall 2011 Linear Algebra Help...

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Unformatted text preview: MATH 304, Fall 2011 Linear Algebra Help sessions: The Math 304/309/311/323 Help Session will be held on MW from 7:30-10:00pm and on TR from 6:30-9:00 p.m., BLOC 160 Homework assignment #2 (due Thursday, September 15) All problems are from Leons book (8th edition). Section 1.3: 1b, 1c, 1e, 1f, Section 1.4: 9 Section 1.5: 10c, 10e, 10h, 11a, 11b MATH 304 Linear Algebra Lecture 5: Inverse matrix (continued). Identity matrix Definition. The identity matrix (or unit matrix ) is a diagonal matrix with all diagonal entries equal to 1. The n n identity matrix is denoted I n or simply I . I 1 = (1), I 2 = parenleftbigg 1 0 0 1 parenrightbigg , I 3 = 1 0 0 0 1 0 0 0 1 . In general, I = 1 0 . . . 0 1 . . . . . . . . . . . . . . . 0 0 . . . 1 . Theorem. Let A be an arbitrary m n matrix. Then I m A = AI n = A . Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted A 1 , such that AA 1 = A 1 A = I . If A 1 exists then the matrix A is called invertible . Otherwise A is called singular . Let A and B be n n matrices. If A is invertible then we can divide B by A : left division: A 1 B, right division: BA 1 . Remark. There is no notation for the matrix division and the notion is not used often. Basic properties of inverse matrices If B = A 1 then A = B 1 . In other words, if A is invertible, so is A 1 , and A = ( A 1 ) 1 . The inverse matrix (if it exists) is unique. Moreover, if AB = CA = I for some n n matrices B and C , then B = C = A 1 . Indeed, B = IB = ( CA ) B = C ( AB ) = CI = C . If n n matrices A and B are invertible, so is AB , and ( AB ) 1 = B 1 A 1 . ( B- 1 A- 1 )( AB ) = B- 1 ( A- 1 A ) B = B- 1 IB = B- 1 B = I , ( AB )( B- 1 A- 1 ) = A ( BB- 1 ) A- 1 = AIA- 1 = AA- 1 = I . Similarly, ( A 1 A 2 . . . A k ) 1 = A 1 k . . . A 1 2 A 1 1 . Inverting diagonal matrices Theorem A diagonal matrix D = diag ( d 1 , . . . , d n ) is invertible if and only if all diagonal entries are nonzero: d i negationslash = 0 for 1 i n . If D is invertible then D 1 = diag ( d 1 1 , . . . , d 1 n ). d 1 . . . d 2 . . . . . . . . . . . . . . . . . . d n 1 = d 1 1 . . . d 1 2 . . . . . . . . . . . . . . . . . . d 1 n Inverting diagonal matrices Theorem A diagonal matrix D = diag ( d 1 , . . . , d n ) is invertible if and only if all diagonal entries are nonzero: d i negationslash = 0 for 1 i n ....
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Lect1-05-web - MATH 304, Fall 2011 Linear Algebra Help...

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