<Text-field style="Heading 1" layout="Heading 1">Exerciese 1 (10p)</Text-field>

Computer Algebra Systems

Academic Semester 2011-2012 Spring

Midterm Test 02

Name:

ETR: e-Mail:

Please send the solved worksheet in e-mail to me.

Please write only into th blocks marked with "Solution".

Evaluation:

You can reach 100 points.

-30 points: 1

31-49 points: 2

50-59 points: 3

60-79 points: 4

80- points: 5

<Text-field style="Heading 1" layout="Heading 1">Exerciese 1 (10p)</Text-field>

Multiply the following matrices:

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and 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

<Text-field style="Heading 2" layout="Heading 2">Solution</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Exerciese 2 (10p)</Text-field>

Compute the following matrix:

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

<Text-field style="Heading 2" layout="Heading 2">Solution</Text-field>

JSFH

<Text-field style="Heading 1" layout="Heading 1">Exerciese 3 (35 p)</Text-field>

We know that 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

Compute the determinant of the matrix:

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

<Text-field style="Heading 2" layout="Heading 2">Solution</Text-field>

JSFH

<Text-field style="Heading 1" layout="Heading 1">Exerciese 4 (10p)</Text-field>

Find integer solutions of the equation 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

<Text-field style="Heading 2" layout="Heading 2">Solution</Text-field>

JSFH

<Text-field style="Heading 1" layout="Heading 1">Exerciese 5 (35p)</Text-field>

Make a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiNkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRjJGK0Yv square matrix which (i,j)-th element is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYyLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUkjbW5HRiQ2JFEiMkYnL0Y5USdub3JtYWxGJy8lK2V4ZWN1dGFibGVHRjcvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnRkEvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiK0YnRkEvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlEvJSlzdHJldGNoeUdGUS8lKnN5bW1ldHJpY0dGUS8lKGxhcmdlb3BHRlEvJS5tb3ZhYmxlbGltaXRzR0ZRLyUnYWNjZW50R0ZRLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGam4tRjA2JS1GLDYlUSJqRidGNUY4RjtGSC1GTDYtUSgmbWludXM7RidGQUZPRlJGVEZWRlhGWkZmbkZobkZbb0YyLUZMNi1RJyZzZG90O0YnRkFGT0ZSRlRGVkZYRlpGZm4vRmluUSYwLjBlbUYnL0Zcb0Zpb0Zfb0Zib0YyLUZMNi1RIn5GJ0ZBRk9GUkZURlZGWEZaRmZuRmhvRmpvLUZMNi1RKiZ1bWludXMwO0YnRkFGT0ZSRlRGVkZYRlpGZm5GaG5GW29GX29GQ0ZFRkE= .

With the LinearAlgebra package (see the HELP):

1. Compute the determinant of this Matrix

2. Compute its characteristic polynomial (see the HELP)

3. Store this polynomial in a variable, compute its degree.

4. Find the roots of this polynomial

5. Factorize it.

6. Plot it (intelligently)

<Text-field style="Heading 2" layout="Heading 2">Solution</Text-field>

JSFH

TTdSMApJNVJUQUJMRV9TQVZFLzM1NjMyODg4WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQmJSJtRzYkIiIiRikmRic2JCIiI0YpJkYnNiQiIiRGKSZGJzYkRilGLCZGJzYkRixGLCZGJzYkRi9GLCZGJzYkRilGLyZGJzYkRixGLyZGJzYkRi9GL0YlTTdSMApJNVJUQUJMRV9TQVZFLzM1NjM4NTI4WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQmJSJtRzYkIiIiRikmRic2JCIiI0YpJkYnNiQiIiRGKSZGJzYkRilGLCZGJzYkRixGLCZGJzYkRi9GLCZGJzYkRilGLyZGJzYkRixGLyZGJzYkRi9GL0Yl