What is the determinant of a matrix?

The determinant of a matrix is a scalar value that can be calculated using various methods.

The determinant of a matrix is a scalar value that can be calculated from the elements of the matrix. It is denoted by det(A) or |A|. For a 2x2 matrix A = [a b; c d], the determinant is given by det(A) = ad - bc. For a 3x3 matrix A = [a b c; d e f; g h i], the determinant can be calculated using the Laplace expansion method or the Sarrus rule. The Laplace expansion method involves expanding the determinant along a row or column and recursively calculating the determinants of the submatrices. For example, to calculate the determinant of a 3x3 matrix A using the Laplace expansion along the first row, we have det(A) = a(det(B)) - b(det(C)) + c(det(D)), where B, C, and D are 2x2 matrices obtained by deleting the first row and the corresponding column from A. The Sarrus rule is a mnemonic for calculating the determinant of a 3x3 matrix using a diagonal pattern.

The determinant of a matrix has several important properties. It is zero if and only if the matrix is singular, i.e., it has no inverse. It is also a linear function of the rows or columns of the matrix, and it changes sign if two rows or columns are interchanged. Moreover, the determinant of a product of matrices is equal to the product of their determinants, i.e., det(AB) = det(A)det(B). This property is useful for calculating the determinant of a matrix by reducing it to a product of simpler matrices. Finally, the determinant of a matrix can be used to solve systems of linear equations, to find the area or volume of geometric objects, and to determine whether a linear transformation preserves orientation.