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

(1)

If the determinant of system (1)

(2)

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

where Dk is the determinant obtained from the determinant of system D by replacing the elements a1k, a2k, ..., ank of the kth column by the corresponding free terms b1, b2, ..., bn or

(3)

where Aik is a cofactor of the element aik 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.

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