As another poster has also said, the determinant of a matrix provides 2 very important pieces of information about the associated linear transformation of the space.
The sign of the determinant tells you whether the linear transformation includes a mirror reflection of the space, or not.
The absolute value of the determinant tells you whether the linear transformation preserves the (multi-dimensional) volume (i.e. it is an isochoric transformation, which changes the shape without changing the volume), or it is an expansion of the space or a compression of the space, depending on whether the absolute value of the determinant is 1, greater than 1 or less than 1.
To understand what a certain linear transformation does, one usually decomposes it in several kinds of simpler transformations (by some factorization of the matrix), i.e. rotations and reflections that preserve both size and shape (i.e. they are isometric transformations), isochoric transformations that preserve volume but not shape, and similitude transformations (with the scale factor computed from the absolute value of the determinant), which preserve shape, but not volume. The determinant provides 2 of these simpler partial transformations, the reflection and the similitude transformation.
Suppose you have (let's say) a 3x3 matrix. This is a linear transformation that maps real vectors to real vectors. Now let's say you have a cube as input with volume 1, and you send it into this transformation. The absolute value of the determinant of the matrix tells you what volume the transformed cube will be. The sign tells you if there is a parity reversal or not.
Here are three reasons you want to be able to calculate the volume change for arbitrary parallelpipeds:
- If det M = 0, then M is not invertible. Knowing this is useful for all kinds of reasons. It means you cannot solve an equation like Mx = b by taking the inverse ("dividing") on both sides, x = M \ b. It means you can find the eigenvalues of a matrix by rearranging Mx = λx <--> (M-λI)x = 0 <--> det M-λI = 0, which is a polynomial equation.
- Rotations are volume-preserving, so the rotation group can be expressed as the matrices where det M = 1 (well, the component connected to the identity). This is useful for theoretical physics, where they're playing around with such groups and need representations they can do things with.
- In information theory, the differential entropy (or average amount of bits it takes to describe a particular point in a continuous probability distribution) increases if you spread out the distribution, and decreases if you squeeze it together by exactly log |det M| for a linear transformation. A nonlinear transformation can be linearized with its gradient. This is useful for image compression (and thus generation) with normalizing flow neural networks.
Form a quadratic equation to solve for the eigenvalues x of a 2x2 matrix (|A - xI| = 0). The inverse of a matrix can be calculated as the classical adjugate multiplied by the reciprocal of the determinant. Use Cramer's Rule to solve a system of linear equations by computing determinants. Reason that if x is an eigenvalue of A then A - xI has a non-trivial nullspace (using the mnemonic |A - xI| = 0).
