Homework_2

Homework_2 - CSC 302 Introduction to Numerical Methods...

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Unformatted text preview: CSC 302 Introduction to Numerical Methods Homework #2 Question 1. Compute the determinants of the following matrices. A = 1 2 3 1 0 1 5 0 1 , B = 1- 2 3 1- 1 5 1- Answer: For matrix A it is easiest to expand along the second column. We obtain det( A ) = a 12 cofactor ( A 12 ) = 2 (- 1) 3 1 1 5 1 = 2 (- 1) 3 (- 4) = 8 . We shall also expand B along its second column. det( B ) = b 12 cofactor ( B 12 ) + b 22 cofactor ( B 22 ) = 2 (- 1) 3 1 1 5 1- + (- ) (- 1) 4 1- 3 5 1- = 2 (- 1) 3 [(1- )- 5] + (- ) (- 1) 4 [(1- )(1- )- 15] = 2( + 4)- 2- 2 - 14 =- 3 + 2 2 + 16 + 8 . 1 Question 2. Let x < n and consider the function f ( x ) = k x k 1 + k x k where and are two nonzero scalars. Check this function against the three conditions necessary for it to be a vector norm and determine if in fact it is a norm. Answer: (a) Definite/Positivity: f ( x ) = || x || 1 + || x || is not necessarily For example, =- 1, =- 1, then f ( x ) < 0. So, fails positivity: will not be a norm. (b) Homogeneity: f ( x ) = || x || 1 + || x || = | ||| x || 1 + | ||| x || = | | ( || x || 1 + || x || ) = | | f ( x ) So, it passes homogeneity. (c) Triangle inequality: f ( x + y ) = || x + y || 1 + || x + y || ( || x || 1 + || y || 1 ) + ( || x || + || y || ) f ( x + y ) ( || x || 1 + || x || ) + ( || y || 1 + || y || ) f ( x + y ) f ( x ) + f ( y ) So, it passes triangle inequality....
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Homework_2 - CSC 302 Introduction to Numerical Methods...

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