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 Leon’s 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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