We all know that a matrix can have m rows and n columns.

However, in the very special case of m = n = 1 we have a one-by-one matrix.

Question is, what is the true nature of a one-by-one matrix?

Is it conceptually the same thing as the number in it?

Or is it a different creature?

For example can I write:

Quote:

[5] = 5

[k + r] = k + r

But, it looks like with matrices, the one-by-one matrix is the same as the element inside it.

Is this true?

If yes, why so different from sets?

If no, why?

I would appreciate it if anybody could shed some light on this issue.

Thanks. ]]>

Let's say we have a function as follows:

f(x) = 3x^{2} - 2x + 7

The goal is to find f '(5x).

I try to calculate that in two different ways:

Method 1:

f '(x) = 6x - 2

f '(5x) = 6(5x) - 2

__f '(5x) = 30x - 2__

f '(5x) = 6(5x) - 2

Method 2:

f(x) = 3x^{2} - 2x + 7

f(5x) = 3(5x)^{2} - 2(5x) + 7

f(5x) = 75x^{2} - 10x + 7

__f '(5x) = 150x - 10__

f(5x) = 3(5x)

f(5x) = 75x

That is two completely different results for f '(5x) !!!!!!

How can that be?

If you solve a problem in two (or many) different ways, the result should be the same.

So, why are the results different in this case?

Which one is correct?

And what is wrong in the line of reasoning that leads to the other?

Please help.

Thanks. ]]>