Please, I'm studying Matrix Algebra and am faced with the hard formulas for calculating a determinant of a Matrix. However, I really don't understand what does a determinant abstractly mean. I tried to search for a conceptual definition for the determinant in order to be able to understand and accept the idea of the determinant. But all my resources provide a functional definition.

For example; in Wikipedia, they say, it's a useful value that can be calculated by a square matrix. In an abstract algebra text, the author defines it via Leibniz's formula. In "The Theory of Matrices", prof. Matcher doesn't provide a definition at all.

What I'm looking for is: What does the determinant really mean? Why did they need to calculate it? Why did they calculate it that way and not in another way? Does the determinant provide any information about the equations that the matrix represents?

I hope I could explain what I mean... Thanks...

up vote 1 down vote accepted

Let’s try and motivate the determinant first by thinking of (oriented) volumes. If we want the determinant of a matrix to represent the volume of the parallelopiped spanned by its columns, we’d want the following properties:

1) Multiplying a row by a scalar multiplies the determinant by that number. We want this so that any stretches affect my volume appropriately. 2) Adding a multiple of one row to another doesn’t affect the determinant. We want this because intuitively we know that shearing (which is basically what adding a multiple of one row to another) doesn’t change the volume. 3) Switching two rows changes the sign. We need this if we want oriented volume. This is a little abstract. 4) The determinant of the identity is 1. We want this because we know that the volume of a unit box is 1.

Turns out we can encapsulate these properties by saying that we want a multilinear (i.e. linear in each entry) alternating (i.e. switches sign if you swap entries once) function from $n$ vectors in $\mathbb{R}^n$ to $\mathbb{R}$ such that its value on the identity is 1. Interestingly, there is only one such function, and that is what we call the determinant. This method defines the determinant as an alternating tensor.

Alternative formulations include motivating it via group theory, by taking a sum over the symmetric group on $n$ elements of appropriate terms. You can also define it via a recursive formula which is what we tend to do in intro linear algebra courses. All these formulations are equivalent, which hints toward the fact that’s it’s something essential.

  • Thanks! I really appreciate your simple explanation :) – Abdu Magdy Nov 13 '17 at 8:31

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