To understand how determinants are evaluated, let us go through the process step by step, starting from the simplest 1×1 matrix and gradually moving to more complex and special cases.
Determinants are the scalar quantities obtained by the sum of products of the elements of a square matrix and their cofactors according to a prescribed rule. They help to find the adjoint, inverse of a matrix.
Determinants are mathematical objects that are very useful in the analysis and solution of systems of linear equations. As shown by Cramer's rule, a nonhomogeneous system of linear equations has a unique solution iff the determinant of the system's matrix is nonzero (i.e., the matrix is nonsingular).
Determinants are of use in ascertaining whether a system of n equations in n unknowns has a solution. If B is an n × 1 vector and the determinant of A is nonzero, the system of equations AX = B always has a solution.
Learn some ways to eyeball a matrix with zero determinant, and how to compute determinants of upper- and lower-triangular matrices. Learn the basic properties of the determinant, and how to apply them.
There are a number of methods used to find the determinants of larger matrices. Cofactor expansion, sometimes called the Laplace expansion, gives us a formula that can be used to find the determinant of a matrix A from the determinants of its submatrices.
Determinants are derived from matrices, and matrices come from systems of linear equations. So determinants are a part of mathematics called "linear algebra" or " matrix algebra".