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Unformatted text preview: Math 330 Exam 2, March 2, 2009 No calculators are allowed on the following. Be sure to show all your work, especially any row operations. Matrix Operations  20 points Consider the following matrices: A = bracketleftBigg 1 2 3 1 1 bracketrightBigg ,B = 1 5 1 0 10 Compute each of the following and answer the indicated questions (5 points each part) (a) What is A T ? What is A T + B ? SOLUTION: bracketleftBigg 1 2 3 1 1 bracketrightBigg T + 1 5 1 0 10 = 1 2 1 3 1 + 1 5 1 0 10 = 2 5 3 1 3 9 (b) Find a matrix C so that C + 3 A is the appropriate sized matrix with all entries zero. SOLUTION: C + 3 A = 0 means that C = 3 A . So C = 3 A = 3 bracketleftBigg 1 2 3 1 1 bracketrightBigg = bracketleftBigg 3 6 9 3 3 bracketrightBigg (c) Calculate BA . Does AB = BA ? Explain and/or calculate. SOLUTION: 1 5 1 0 10 bracketleftBigg 1 2 3 1 1 bracketrightBigg = 1 3 2 1 2 3 10 10 Since AB is a 2 × 2 matrix and BA is a 3 × 3 matrix they cannot be equal. (d) Specify any matrix D (with all entries nonzero), so that DB = bracketleftBigg 0 0 0 0 bracketrightBigg . We should make D a 2 × 3 matrix so that DB is 2 × 2. The two columns of B give weights for linear combinations of the columns of D which produce the zero vector. Therefore, with D = [ vector d 1 vector d 2 vector d 3 ] then we need (from the first column of B ) that vector d 1 + vector d 2 = vector 0. From the second column of B we need that 5 vector d 1 + 10 vector d 3 = vector 0. There are many possibilities, one of the simpler ones D = bracketleftBigg 2 2 1 2 2 1 bracketrightBigg 1 Inverse Matrices  20 points Find the inverses (or explain why these do not exist) of each of the following matrices: (10 pts each) (a) A =  1 0 0 0 1 0 0 0 0 5 0 1 0 SOLUTION:  1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 5 0 0 1 0 0 1 0 0 0 0 1 R 1 ← R 1 & R 3 ← R 3 5 = ⇒ 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 5 0 0 1 0 0 0 1...
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This note was uploaded on 09/19/2009 for the course MATH 330 taught by Professor Johnson,j during the Spring '08 term at Nevada.
 Spring '08
 Johnson,J
 Math, Linear Algebra, Algebra, Matrices, Matrix Operations

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