For more complicated matrices, the Laplace formula (cofactor expansion), Gaussian elimination or other algorithms must be used to calculate the determinant. Some useful decomposition methods include QR, LU and Cholesky decomposition. The determinant of the product of matrices is equal to the product of determinants of those matrices, so it may be beneficial to decompose a matrix into simpler matrices, calculate the individual determinants, then multiply the results. The prompts for the functions are quite similar. DSolve is used when the user wishes to find the general function or functions which solve the differential equation, and NDSolve is used when the user has an initial condition. For a 2-by-2 matrix, the determinant is calculated by subtracting the reverse diagonal from the main diagonal, which is known as the Leibniz formula. Mathematica features two functions for solving ODEs: DSolve and NDSolve. Some matrices, such as diagonal or triangular matrices, can have their determinants computed by taking the product of the elements on the main diagonal. There are many methods used for computing the determinant. Geometrically, the determinant represents the signed area of the parallelogram formed by the column vectors taken as Cartesian coordinates. A system of linear equations can be solved by creating a matrix out of the coefficients and taking the determinant this method is called Cramer's rule, and can only be used when the determinant is not equal to 0. A determinant of 0 implies that the matrix is singular, and thus not invertible. The value of the determinant has many implications for the matrix. Knowledgebase about determinants A determinant is a property of a square matrix.
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