Determinants
Determinants are important concepts in linear algebra. It is used in solving linear systems of equations, testing the invertibility of matrices, finding the eigenvalues & eigenvectors and more. It is only defined for square matrices of the form \(A^{n \times{n}}\) and is usually denoted by \(\det(A)\) or \(|A|\).
Definition: The determinant of a square matrix \(A \in \mathbb{R}^{n \times{n}}\) is a function (of it's entries) that maps \(A\) onto a real number.
Geometrically speaking determinants represent the factor by which a region of space is scaled by a linear transformation (represented by a matrix). For a two dimensional linear transformation this would be the factor by which a plane of unit area gets scaled to ie., either enlarged or shrunk after applying the transformation.
As you can see above the area is scaled by a factor of 6 on applying the linear transformation represented by the matrix. Similarly for 3 dimensions the determinants provide the factor by which volumes are scaled or simply the volume of a 3-dimesional parallelepiped (since we're a scaling up or down from a region of unit volume).
This intuition extends to higher dimensions. Generally we could say that the determinant of a matrix \(A\) represents the signed volume of the n-dimensional parallelepiped formed by the column vectors of \(A\). The sign of the determinant indicates the orientation of the basis vectors in space. For example in the previous set of images a -ve determinant would swap the positions of the red and the green basis vectors much similar to flipping a sheet of paper. And a determinant of value 0 would mean all of space has been squished to form a region of lower dimension, a plane in case of 3-d tranformations and a line in case of 2-d tranformations.