Find Determinant 2

[harpazo]

Let D = determinant

Find D given the following:

\begin{bmatrix}2&5\\

8&1\end{bmatrix}

D = 2(1) - 8(5)

D = 2 - 40

D = - 38

Can the determinant be negative? Sullivan provides a fuzzy definition for the determinant. Perhaps, I need someone here to clarify this word. In the previous post, D = 0. Here, D = - 38. What exactly does that mean? I know D can also be a positive number. When I find the determinant in given problems, what exactly have I found?

 

[Dr.Peterson]

A determinant can have any value you can imagine.

To really explain what the determinant means, you would need to be learning linear algebra. That's one reason some people would consider it inappropriate to be learning about them outside of a linear algebra course. It's sort of, "We're going to use this idea that you are not really ready for, so just trust us and do what we say." Math people don't like doing that.

So, for your purposes, a determinant is just a number you can calculate, that's useful in Cramer's Rule. The sign probably doesn't affect anything you'd do with it; whether it is zero definitely does.

As for what it really is, maybe start here: https://en.wikipedia.org/wiki/Determinant

A nice quote from there, if you want a brief statement of what a determinant means:

Geometrically, it can be viewed as the volume scaling factor of the linear transformation described by the matrix. This is also the signed volume of the n-dimensional parallelepiped spanned by the column or row vectors of the matrix. The determinant is positive or negative according to whether the linear mapping preserves or reverses the orientation of n-space.​

There are some very interesting details hidden in all that! But none of them are going to be in your chapter.

 

[harpazo]

A determinant can have any value you can imagine.

To really explain what the determinant means, you would need to be learning linear algebra. That's one reason some people would consider it inappropriate to be learning about them outside of a linear algebra course. It's sort of, "We're going to use this idea that you are not really ready for, so just trust us and do what we say." Math people don't like doing that.

So, for your purposes, a determinant is just a number you can calculate, that's useful in Cramer's Rule. The sign probably doesn't affect anything you'd do with it; whether it is zero definitely does.

As for what it really is, maybe start here: https://en.wikipedia.org/wiki/Determinant

A nice quote from there, if you want a brief statement of what a determinant means:

Geometrically, it can be viewed as the volume scaling factor of the linear transformation described by the matrix. This is also the signed volume of the n-dimensional parallelepiped spanned by the column or row vectors of the matrix. The determinant is positive or negative according to whether the linear mapping preserves or reverses the orientation of n-space.​

There are some very interesting details hidden in all that! But none of them are going to be in your chapter.

Interesting. Thanks for the link provided.