In linear algebra, the determinant is a scalar value that can be calculated from the elements of a square matrix. It can be used to perform certain operations on the matrix, such as finding the inverse of a matrix, testing whether a matrix is singular (non-invertible), and finding the volume of the parallelepiped spanned by its columns.

The determinant of a matrix is denoted as det(A), or |A|, and it is calculated by a specific set of rules that depend on the size of the matrix. For a 2x2 matrix, the determinant is simply the difference between the product of the diagonal elements and the product of the off-diagonal elements:

|A| = a11a22 - a12a21

For a 3x3 matrix, the determinant is more complex and is calculated using a set of rules known as the "determinant expansion by minors."

Determinants can be used to solve systems of linear equations, compute matrix inverses, and find matrix rank. They are also used in calculus to compute triple integrals and in physics to calculate the absolute value of the Jacobian matrix in transformations of coordinates.

The determinant of a matrix represents the scaling factor of the transformation represented by the matrix. For example, if A is a 2x2 matrix that represents a rotation in the plane, then the determinant of A is the scaling factor of the transformation. If the determinant is 1, then the transformation is an isometry, meaning that it preserves distances. If the determinant is -1, then the transformation is an isometry that also reflects the plane. If the determinant is 0, then the matrix is singular and does not represent a valid transformation.

In general, the determinant of a matrix represents the "size" of the transformation it represents. For example, if A is a 3x3 matrix that represents a linear transformation in 3D space, then the determinant of A is the scaling factor of the transformation. It tells you how much the transformation "stretches" or "compresses" space in each dimension. The determinant can be positive, negative, or zero. If the determinant is positive, then the transformation expands space in some dimensions and contracts it in others. If the determinant is negative, then the transformation reflects space across some plane, in addition to expanding or contracting it. If the determinant is zero, then the transformation is singular and does not have an inverse.

In linear algebra, an eigenvector of a linear transformation is a non-zero vector that, when the transformation is applied to it, changes only in scale (not direction). Formally, if T is a linear transformation and v is a vector that is not the zero vector, then v is an eigenvector of T if T(v) is a scalar multiple of v. The scalar value that is used to scale v is called the eigenvalue of T associated with v.

Eigenvectors and eigenvalues are used to diagonalize matrices, which can simplify many calculations. For example, if A is a square matrix and v is an eigenvector of A, then Av is simply a scalar multiple of v. This means that if we express A as a linear combination of eigenvectors, the resulting matrix will be diagonal, which makes it much easier to compute powers of A or to find the inverse of A.

Eigenvectors and eigenvalues also have important applications in physics and engineering, where they are used to analyze the behavior of systems that can be described by matrices. For example, eigenvectors and eigenvalues are used to solve differential equations, to analyze the stability of equilibrium points in dynamical systems, and to diagonalize the mass matrix in quantum mechanics.