Determinant identities

In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix A is denoted det(A), det A, or |A|. Geometrically, it can be viewed as the volume scaling factor of the linear transformation described by the matrix. This is also the signed volume of the n-dimensional parallelepiped spanned by the column or row vectors of the matrix. The determinant is positive or negative according to whether the linear mapping preserves or reverses the orientation of n-space. In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix A is denoted det(A), det A, or |A|. Geometrically, it can be viewed as the volume scaling factor of the linear transformation described by the matrix. This is also the signed volume of the n-dimensional parallelepiped spanned by the column or row vectors of the matrix. The determinant is positive or negative according to whether the linear mapping preserves or reverses the orientation of n-space. In the case of a 2 × 2 matrix the determinant may be defined as: Similarly, for a 3 × 3 matrix A, its determinant is: Each determinant of a 2 × 2 matrix in this equation is called a 'minor' of the matrix A. This procedure can be extended to give a recursive definition for the determinant of an n × n matrix, the minor expansion formula. Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficients in a system of linear equations, and the determinant can be used to solve those equations, although other methods of solution are much more computationally efficient. In linear algebra, a matrix (with entries in a field) is invertible (also called nonsingular) if and only if its determinant is non-zero, and correspondingly the matrix is singular if and only if its determinant is zero. This leads to the use of determinants in defining the characteristic polynomial of a matrix, whose roots are the eigenvalues. In analytic geometry, determinants express the signed n-dimensional volumes of n-dimensional parallelepipeds. This leads to the use of determinants in calculus, the Jacobian determinant in the change of variables rule for integrals of functions of several variables. Determinants appear frequently in algebraic identities such as the Vandermonde identity. Determinants possess many algebraic properties, including that the determinant of a product of matrices is equal to the product of determinants. Special types of matrices have special determinants; for example, the determinant of an orthogonal matrix is always plus or minus one, and the determinant of a complex Hermitian matrix is always real. If an n × n real matrix A is written in terms of its column vectors A = [ a 1 a 2 ⋯ a n ] {displaystyle A=} then This means that A {displaystyle A} maps the unit n-cube to the n-dimensional parallelotope defined by the vectors a 1 , a 2 , … , a n , {displaystyle mathbf {a} _{1},mathbf {a} _{2},ldots ,mathbf {a} _{n},} the region P = { c 1 a 1 + ⋯ + c n a n ∣ 0 ≤ c i ≤ 1   ∀ i } . {displaystyle P=left{c_{1}mathbf {a} _{1}+cdots +c_{n}mathbf {a} _{n}mid 0leq c_{i}leq 1 forall i ight}.} The determinant gives the signed n-dimensional volume of this parallelotope, det ( A ) = ± vol ( P ) , {displaystyle det(A)=pm { ext{vol}}(P),} and hence describes more generally the n-dimensional volume scaling factor of the linear transformation produced by A. (The sign shows whether the transformation preserves or reverses orientation.) In particular, if the determinant is zero, then this parallelotope has volume zero and is not fully n-dimensional, which indicates that the dimension of the image of A is less than n. This means that A produces a linear transformation which is neither onto nor one-to-one, and so is not invertible.