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What's an intuitive way to think about the determinant?

I just learned the basics of matrices. Then I came across the magical formula $$\det(AB)=\det(A)\det(B)$$

I understand the proof, but I don't really understand why it is true. It seems that we are too lucky. The first step is to prove that $\det(EA)=\det(E)\det(A)$ for elementary matrix $E$. How come we are so lucky that this is true for all three types of elementary matrices?

How did they come up with determinants in the first place? I understand that determinant was originally used to determine the number of solutions of linear equations. But how did they come up with such intricate formula?

Please enlighten me up with intuitive explanations. I'm tired with all the formal proofs.

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marked as duplicate by Qiaochu Yuan Sep 8 '11 at 3:41

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

One day, a Mommy matrix and a Daddy matrix got together... – The Chaz 2.0 Sep 8 '11 at 3:24
The absolute value of the determinant can be thought of as the (hyper)volume of the parallelopiped determined by the images of the standard basis vectors under the action of $B$. In that light, $\det(AB)=\det(A)\det(B)$ "makes sense" in that if $B$ transforms a "unit cube" into a parallelopiped of volume $\det(B)$, and $A$ into one of volume $\det(A)$, you expect $AB$ to transform a "unit cube" into a parallelopiped of volume $\det(A)\det(B)$... – Arturo Magidin Sep 8 '11 at 3:29

There are lots of reasons why the determinant is important. The formula is not one of them.

In particular, a matrix most importantly represents a linear transform. The determinant of the matrix is the volume of the unit box after transformation.

The formula for the determinant happens to be the one that calculates this piece of data.

This makes it clear why $\det AB = \det A \cdot \det B$. The matrix $AB$ represents the transformation by $A$ and $B$ in succession.

The sign of the determinant tells whether or not the ''handedness'' of the space has flipped. The fact that flipping something twice gets you back where you started corresponds to the fact that the product of two negative numbers is positive.

Essentially the reason that determinants work this way isn't that we were ''lucky'', but rather that we wanted them to have this property and so we found a definition gave it to them.

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This Thomas Muir: History of determinants should help

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