# Enlargement factor

• May 15th 2011, 10:30 AM
Stuck Man
Enlargement factor
I think that the absolute value of the determinant is the enlargement factor of a matrix. So if ad-bc is 2.2-0.0 the factor is 4. Is that correct?

A question says "Obtain the 2 × 2 matrix M associated with an enlargement, scale factor 2." I think that the a and d elements must be the square root of 2. Why does the answer say they are 2?
• May 15th 2011, 10:42 AM
HallsofIvy
Quote:

Originally Posted by Stuck Man
I think that the absolute value of the determinant is the enlargement factor of a matrix. So if ad-bc is 2.2-0.0 the factor is 4. Is that correct?

A question says "Obtain the 2 × 2 matrix M associated with an enlargement, scale factor 2." I think that the a and d elements must be the square root of 2. Why does the answer say they are 2?

What, exactly, do you mean by "enlargement factor"? The matrix $\displaystyle \begin{bmatrix}2 & 0 \\ 0 & 2\end{bmatrix}$ multiplys all lengths by 2 and so area by 4.
• May 15th 2011, 10:45 AM
Quacky
Quote:

Originally Posted by Stuck Man
I think that the absolute value of the determinant is the enlargement factor of a matrix. So if ad-bc is 2.2-0.0 the factor is 4. Is that correct?

A question says "Obtain the 2 × 2 matrix M associated with an enlargement, scale factor 2." I think that the a and d elements must be the square root of 2. Why does the answer say they are 2?

You are mistaken.

The determinant of a matrix gives you the area represented by the matrix, not the scale factor of enlargement. An enlargement of scale factor M is defined as $\displaystyle \left(\begin{array}{cc}M&0\\0&M\end{array}\right)$

This can be shown quite easily. Take a general point x,y.

Now apply the transformation:

$\displaystyle \left(\begin{array}{cc}M&0\\0&M\end{array}\right)~ \left(\begin{array}{cc}x\\y\end{array}\right)$

$\displaystyle = \left(\begin{array}{cc}Mx\\My\end{array}\right)$

Edit: Was beaten by HallsofIvy
• May 15th 2011, 10:51 AM
Stuck Man
I think I've found there are subtly different terms. The absolute value of a determinant is the enlargement factor of the area or volume. An enlargement with a scale factor is an enlargement of the distance between points.
• May 15th 2011, 10:58 AM
Stuck Man
Would you say the term "enlargement factor" refers to area/volume if there is nothing else to indicate otherwise?
• May 15th 2011, 11:04 AM
Quacky
Quote:

Originally Posted by Stuck Man
Would you say the term "enlargement factor" refers to area/volume if there is nothing else to indicate otherwise?

No. I would assume it refers to the scale factor of enlargement, which is what I addressed in my previous post.
• May 15th 2011, 11:13 AM
Stuck Man
Since its ambigous I would expect an exam question would make it clear. The question is from an exam.
• May 15th 2011, 11:24 AM
Quacky
Quote:

Originally Posted by Stuck Man
Since its ambigous I would expect an exam question would make it clear. The question is from an exam.

Really? 'Scale factor of enlargement' and 'enlargement factor' seem relatively interchangable to me. That's just personal opinion though. The determinant of a matrix just tells you the area. It doesn't tell you anything at all regarding the enlargement or transformation. Also, they asked you for a matrix, so clearly they didn't want you to calculate a determinant.(Cool)
• May 16th 2011, 02:03 AM
Ackbeet
Quote:

Originally Posted by Stuck Man
Since its ambigous I would expect an exam question would make it clear. The question is from an exam.

A previous exam?
• May 16th 2011, 08:18 AM
Stuck Man
If a matrix is used for a transformation its determinant is the multiplication factor for area.
• May 16th 2011, 08:31 AM
Quacky
Quote:

Originally Posted by Stuck Man
If a matrix is used for a transformation its determinant is the multiplication factor for area.

Ok, I think I see what you mean now, sorry. I still don't agree that it's an ambiguous question though. It specifically asks for a scale factor 2 enlargement - although the wording is slightly questionable, I think the meaning is quite easily interpreted. I can see why it would confuse you, but I have seen other exam questions that have been worded extremely similarly.
• May 16th 2011, 08:47 AM
Stuck Man
I was meaning the term "enlargement factor" in general not in the the question. I also think it usually refers to the scale factor.