with( linalg ):In most linear algebra courses, the algorithm for finding the determinant involves computing the solutoin recursively using a cofactor expansion. The problem withthis approach is that it has complexity O(n!), which is clearly intractable for any nontrivial n.Using the LU decomposition leads to a polynomial time algorithm. Essentially, we use the following theorem:Theorem 3.2.1 (Matrix Computations, Golub and Loan)Real, nxn matrix A has an LU factorization if det(A(1:k,1:k)) != 0 for k = 1:n-1.If an LU factorization exists and A is nonsingular, then the LU factorization is unique and det(A)=u_11..u_nnSo essentially we use gaussian elimination to check if the matrix is nonsingular, and if its not we can use the results of the elimination, the product of the diagonal entries of U, to get the determinant. A := matrix( 3, 3, [3,4,5,1,2,3,3,1,5] );B := LUdecomp( A );det( A );B[1,1] * B[2,2] * B[3,3];