*Cramer's rule* is a simple method of solving n linear equations in n unknowns.

If the *determinant* of system (1)

is not zero, then system (1) has a unique solution

where D_{k} is the determinant obtained from the determinant of system D by replacing the elements a_{1k}, a_{2k}, ..., a_{nk} of the kth column by the corresponding free terms b_{1}, b_{2}, ..., b_{n} or

where A_{ik} is a cofactor of the element a_{ik} of determinant D.

Thus, the solution of a linear system of equations (1) in n unknowns reduces to the calculation of the (n + l)th determinant of order n. The number of operations required to solve the system of equations (1) with the help of Cramer's rule is thus proportional to (n+1). For a sufficiently large n, the solution of system (1) with the use of Cramer's rule is computer-time consuming and presents a considerable challenge to practical calculations of heat transfer problems.

Heat & Mass Transfer, and Fluids Engineering