Difference between revisions of "Selftest: Matrix multiplication with a scalar"

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{{ExerciseNavigation|previous=[[Selftest:Cross product|Cross product]]|article=[[Matrices]]|next=[[Selftest:Unit vector|Unit vector]]}}
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{{ExerciseNavigation|previous=[[Selftest: Minors and cofactors|Minors and cofactors]]|chapter=[[Matrices]]|article=[[Multiplication with a scalar]]|next=[[Selftest: Transpose|Transpose]]}}
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<br/>
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<quiz display=simple>
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{'''Is there any scalar constant <math>c</math>, so that the following equation holds?'''<br/>
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:<math>
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\left[\begin{array}{ccc}1&2&3\\0&2&1\\2&3&0\end{array}\right]\cdot c = \left[\begin{array}{ccc}2&4&6\\0&4&2\\3&6&0\end{array}\right]
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</math><br/>
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| typ="()" }
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- <math>c=0.5</math>
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- <math>c=1</math>
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- <math>c=2</math>
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+ There is no <math>c</math>
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||<math>c=2</math> would be correct for all the green colored components in the result matrix: <math>\left[\begin{array}{ccc}{\color{Green}2}&{\color{Green}4}&{\color{Green}6}\\{\color{Green}0}&{\color{Green}4}&{\color{Green}2}\\{\color{Red}3}&{\color{Green}6}&{\color{Green}0}\end{array}\right]</math>. For the red component <math>c=1.5</math> would be right. So there is no general <math>c</math>, that holds for all the components.
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 +
{'''Is there any scalar constant <math>c</math>, so that the following equation holds?'''<br/>
 +
:<math>
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\left[\begin{array}{ccc}2&0&1\\0&2&3\\1&1&0\end{array}\right]\cdot c = \left[\begin{array}{ccc}4&0&2\\0&4&6\\2&2&0\end{array}\right]
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</math><br/>
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| typ="()" }
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- <math>c=0.5</math>
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- <math>c=1</math>
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+ <math>c=2</math>
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- There is no <math>c</math>
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||If all the components of the left matrix are multiplied by <math>c=2</math>, it results in the right matrix.
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{'''Fill in the correct values of the resulting matrix:'''<br/>
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|typ="{}" }
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<math>\left[\begin{array}{ccc}3&2&1\\2&1&2\\2&3&1\end{array}\right]\cdot 3 =</math>
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<br/>{ 9 _2 } { 6 _2 } { 3 _2 }<br/>{ 6 _2 } { 3 _2 } { 6 _2 }<br/>{ 6 _2 } { 9 _2 } { 3 _2 }
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{'''Is the following equation correct?'''<br/>
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:<math>\left[\begin{array}{ccc}2&0&1\\4&1&3\\2&2&1\end{array}\right]\cdot 3 = \left[\begin{array}{ccc}6&0&3\\12&2&9\\6&6&3\end{array}\right]</math><br/>
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| typ="()" }
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- Yes
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+ No
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||The central component in the resulting matrix is <math>2</math>. But it has to be <math>3</math>. So the equation is not correct.
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{'''Is the following equation correct?'''<br/>
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:<math>
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\left[\begin{array}{ccc}1&0&2\\2&1&3\\3&2&1\end{array}\right]\cdot 2 = \left[\begin{array}{ccc}2&0&4\\4&2&6\\6&4&2\end{array}\right]
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</math><br/>
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| typ="()" }
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+ Yes
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- No
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||The multiplication holds for each of the components.
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</quiz>

Latest revision as of 10:23, 25 September 2014

← Previous exercise: Minors and cofactors Exercises for chapter Matrices | Article: Multiplication with a scalar Next exercise: Transpose →


1. Is there any scalar constant c, so that the following equation holds?

\left[\begin{array}{ccc}1&2&3\\0&2&1\\2&3&0\end{array}\right]\cdot c = \left[\begin{array}{ccc}2&4&6\\0&4&2\\3&6&0\end{array}\right]
c=0.5
c=1
c=2
There is no c
→c=2 would be correct for all the green colored components in the result matrix: \left[\begin{array}{ccc}{\color{Green}2}&{\color{Green}4}&{\color{Green}6}\\{\color{Green}0}&{\color{Green}4}&{\color{Green}2}\\{\color{Red}3}&{\color{Green}6}&{\color{Green}0}\end{array}\right]. For the red component c=1.5 would be right. So there is no general c, that holds for all the components.

2. Is there any scalar constant c, so that the following equation holds?

\left[\begin{array}{ccc}2&0&1\\0&2&3\\1&1&0\end{array}\right]\cdot c = \left[\begin{array}{ccc}4&0&2\\0&4&6\\2&2&0\end{array}\right]
c=0.5
c=1
c=2
There is no c
→If all the components of the left matrix are multiplied by c=2, it results in the right matrix.

3. Fill in the correct values of the resulting matrix:

\left[\begin{array}{ccc}3&2&1\\2&1&2\\2&3&1\end{array}\right]\cdot 3 =




4. Is the following equation correct?

\left[\begin{array}{ccc}2&0&1\\4&1&3\\2&2&1\end{array}\right]\cdot 3 = \left[\begin{array}{ccc}6&0&3\\12&2&9\\6&6&3\end{array}\right]
Yes
No
→The central component in the resulting matrix is 2. But it has to be 3. So the equation is not correct.

5. Is the following equation correct?

\left[\begin{array}{ccc}1&0&2\\2&1&3\\3&2&1\end{array}\right]\cdot 2 = \left[\begin{array}{ccc}2&0&4\\4&2&6\\6&4&2\end{array}\right]
Yes
No
→The multiplication holds for each of the components.

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