Cramer s rule

By Cramer s rule, each solution of Eqs. (2-44) is given as the ratio of determinants... 

Solving the normal equations by Cramer s rule leads to the solution set in determinantal fomi... 

IB I/IAI, where Bi is the determinant obtained from A by replacing its /cth column by Z i, ho,, h . This techniqiie is called Cramer s rule. It requires more labor than the method of elimination and should not be used for computations. 

The foregoing result along with Equation (A.23) is known as Cramer s rule. If Y in Equation (A.25) is zero, then the system of equations... 

We have written the equations in a form that lets us apply Cramer s rule. The result is... 

Equation (125) applies for all values of the index k — 1,2,..., m. It is a set of m simultaneous, homogeneous, linear equations for the unknown values of the coefficients c . Following Cramer s rule (Section 7.8), a nontrivial solution exists only if the determinant of the coefficients vanishes. Thus, the secular determinant takes the form... 

If the unit matrix E is of order n, Eq. (67) represents a system of n homogeneous, linear equations in n unknowns. They are usually referred to as the secular equations. According to Cramer s rule [see (iii) of Section 7.8], nontrivial solutions exist only if the determinant of the coefficients vanishes. Thus, for the solutions of physical interest,... 

This result is a system of simultaneous linear, homogeneous equations for the coefficients, cu. Cramer s rule states that a nontrivial solution exists only if... 

To summarize what is referred to as Cramer s rule, we can use the following general expressions given a system of two equations (6-13a and 6-13b) in two unknowns such that... 

Gaussian elimination is a very efficient method for solving n equations in n unknowns, and this algorithm is readily available in many software packages. For solution of linear equations, this method is preferred computationally over the use of the matrix inverse. For hand calculations, Cramer s rule is also popular. 

It is straightforward to show by applying Cramer s rule recursively to chains of increasing N that the determinant of matrix [A] is 1, with det[A] = —1 for N... 

The use of the KrOnecker delta to define the overlap matrix ensures that E appears only in the diagonal elements of the determinant. Since this is a 3 x 3 determinant, it may be expanded using Cramer s rule as... 

Eqna. (A.4-2.4) give the solution of any set of n equations in n variables the method is known as Cramer s rule. 

Let Rk be the determinant obtained by replacement of the elements of column k in det(a,-,-) with bvb2,...,bn. Cramer s rule states that for inhomogeneous equations, the unknown xk is given by... 

A set of n linear homogeneous equations in n unknowns always has the solution x, — x2 = xn = 0, which is the trivial solution. Suppose the coefficient determinant det(aiy) is not equal to zero for a set of linear homogeneous equations we can then use Cramer s rule (1.82). Since the equations are homogeneous, the determinant Rk will have a column of zeros, and will equal zero hence xt = x2= =xn = 0, and we have only the trivial solution. Thus for a nontrivial solution of the homogeneous equations to exist, we must have det(o,y) = 0. This condition can also be shown to be sufficient to insure the existence of a nontrivial solution. A system of n simultaneous, linear, homogeneous equations in n unknowns has a nontrivial solution if and only if the determinant of the coefficients equals zero. 

If xx = dx,x2 = d2,...,xn = dn is a solution of a set of n linear homogeneous equations, then xx — cdlyx2 = cd2,...,xn = cdn (where c is any constant) is readily seen to be a solution also. Since we have an arbitrary constant in the solution, to solve a set of linear homogeneous equations, we set one of the unknowns (e.g., j ) equal to an arbitrary constant (x, = c), thereby converting the equations to n equations in n — 1 unknowns we discard one equation and proceed to Solve the set of n — 1 inhomogeneous equations in n — 1 unknowns. (The most efficient way to do this is by successive elimination of unknowns, rather than by Cramer s rule.) In quantum mechanics, the arbitrary constant c is usually evaluated by normalization. 

There are four methods for solving systems of linear equations. Cramer s rule and computing the inverse matrix of A are inefficient and produce inaccurate solutions. These methods must be absolutely avoided. Direct methods are convenient for stored matrices, i.e. matrices having only a few zero elements, whereas iterative methods generally work better for sparse matrices, i.e. matrices having only a few non-zero elements (e.g. band matrices). Special procedures are used to store and fetch sparse matrices, in order to save memory allocations and computer time. 

A linear matrix equation can be solved by the application of Cramer s rule. Assuming the determinant A of the matrix Z is non-zero, the solution of the current can be expressed as... 

Note indices 0 and k indicate A and P, respectively.) With Cramer s rule one finds ... 

The method of determinants (Cramer s Rule) can be used to solve the equations for each model compartment. However, we will focus only on the solution... 

According to Cramer s rule, a system of simultaneous linear equations has a unique solution if the determinant D of the coefficients is non-zero. 

Following the Cramer s rule procedures described in Chapter 9, we construct the determinant to determine Co " " concentration shown in Figure 21-7. 

---