Microsoft Word - ECON2015 Tutorial Set 2 Final XXXXXXXXXXdocx
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ECON2015: Mathematics for Economics I
Tutorial Set 2 (Units 3 to 4) - (contributes 10%)
Question 1 (30 marks)
a) Find the rank of
Ensure that you explain the steps taken. (10 marks)
) Find the inverse of using Elementary Row Operations
Ensure that you explain the steps taken. (20 marks)
Question 2 (25 marks)
a) Let ? = $
? 1 1
1 ? −1
−1 1 ?
(
(i) Compute the value of ???(?). (4 marks)
(ii) For what values of x will the matrix be singular? (2 marks)
) Write down the quadratic form that co
esponds to the following matrix.
$
3 2 4
2 −1 −3
4 −3 1
(
(3 marks)
c) Examine whether the following quadratic forms are positive definite, negative
definite or indefinite.
(i) ?(?) = 3?! + 2?? + 5?? + 7?! − 2?! (3 marks)
(ii) ?(?) = 2?! − 4?! + 10?? − 7?? − 11?? + 3?! (3 marks)
A=
1 −1 2 3
2 6 −8 1
5 3 −2 10
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
A=
1 2 3
0 1 −1
1 2 1
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
Question 1A Marking Scheme
• Explanation of each step – 5 marks
• Workings – 5 marks
Question 1B Marking Scheme
• Explanation of each step – 5 marks
• Workings – 15 marks
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d) Determine whether or not the following matrix is orthogonal. Justify your
esponse. (4 marks)
? =
1
√6
;
√3 1 √2
0 −2 √2
−√3 1 √2
e) Prove that the quadratic equation ?! − 3? = −2? holds true for
? = $
5 −6 −6
−1 4 2
3 −6 −4
(
(6 marks)