133 assignment2

# 133 assignment2 - Show that A is not invertible Hint...

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MATHEMATICS 133 ASSIGNMENT 2 Due in class on October 26 (for Loveys and Anderson’s sections) and on October 25 (for Clay, Shahabi and Kelome’s sections). Instructions: Show all work and justify answers (even where not explicitly requested). Marks may be deducted for lack of neatness (print if necessary). The assignment mark may be based on a randomly selected problem or problems instead of the whole assign- ment. Therefore be sure to solve each problem. In BLOCK CAPITALS, write your INSTRUCTOR’S LAST NAME, your LAST NAME , and your ID number, in the top right corner . 1. Prove or disprove the following statements; (a) if A is invertible then adj ( A ) is invertible (b) if A is invertible then A 2 is invertible (c) if A has a zero entry on the diagonal then A is not invertible. (d) if A is not invertible then for every matrix B , AB is not invertible. (e) if A is a nonzero 2 × 2 matrix such that A 2 + A = 0 then A is invertible. 2. Let A be a 3 × 3 matrix such that the sum of the entries of each row is equal to zero.
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Unformatted text preview: Show that A is not invertible. ( Hint : construct a 3 × 3 invertible matrix B such that det( AB ) = 0). 3. Let r 1 , r 2 and r 3 be the rows of a 3 × 3 matrix A = r 1 r 2 r 3 . Let B = 2 r 1 + 3 r 2 r 3 + 2 r 1 r 2-r 1-3 r 3 If det( A ) = 2, ﬁnd det( B ). 4. Consider the matrix A = 1-2 2-2 1 2-2 3 (a) Show that λ = 1 is an eigenvalue of A . (b) Is A diagonalizable? 5. Prove or disprove (a) If λ is an eigenvalue of A , then λ is also an eigenvalue of A T . (b) If A 2 + A = 0, then λ = 1 cannot be an eigenvalue of A . (c) A 3 × 3 matrix cannot have four distinct eigenvalues. (d) An invertible matrix A can have λ = 0 as eigenvalue. (e) if λ is an eigenvalue of an invertible matrix A , then 1 λ is an eigenvalue of A-1 ....
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## This note was uploaded on 11/29/2010 for the course MATH 222 taught by Professor Karlpeterrussell during the Spring '08 term at McGill.

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