"A numerical value obtained from a square matrix of the coefficients of unknown variables enclosed by two bars by the process of diagonal expansion to tell upon a given algebraic system."
System of Equations: The determinant can be used to determine whether a system of linear equations has a unique solution, no solution, or infinitely many solutions. If the determinant is zero, the matrix is singular, meaning the system has no unique solution.
Inversion of Matrices: A non-zero determinant is a necessary condition for a square matrix to have an inverse. If the determinant is zero, the matrix is non-invertible.
Characteristics of Determinants:
Uniqueness: The determinant is uniquely defined for any square matrix, and it remains consistent regardless of the method used to calculate it.
Linear Property: The determinant is linear in each row and each column separately. This means that if you multiply one row (or column) by a scalar, the determinant is multiplied by that scalar.
Row or Column Interchange: If two rows (or two columns) of a matrix are swapped, the determinant of the matrix changes its sign.
Additive Property: The determinant of the sum of two matrices is not equal to the sum of their determinants. However, if you add a multiple of one row (or column) to another row (or column), the determinant remains unchanged.
Triangular Matrix: The determinant of a triangular matrix (upper or lower triangular) is the product of the elements on the main diagonal.
Zero Determinant: If a matrix has a row or column of zeros, or if any two rows or columns are identical, the determinant is zero.
Multiplicative Property: The determinant of the product of two matrices is the product of their determinants, i.e., det(AB) = det(A) * det(B).
Cofactor Expansion: The determinant of a matrix can be calculated using cofactor expansion along any row or column, which expresses the determinant as a sum of products involving minors and cofactors.
Cofactor of an element is the product of (-1)^i+j and the minor of concerned element.
Minor of an element of a determinant is the subsquare determinant of the given determinant along which the particular element does not exist.
