MATH 330 Exam 2 Solution

MATH 330 Exam 2 Solution - Math 330 Exam 2, March 2, 2009...

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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 non-zero), 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.

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MATH 330 Exam 2 Solution - Math 330 Exam 2, March 2, 2009...

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