 # Use Cramer`s Rule

According to Cramer`s rule, the system has an infinite number of solutions. We will only extend the same process of Cramer`s rule for 2 equations also for a 3×3 system of equations. Here are the steps to solve this system of 3×3 equations in three variables x, y and z by applying Cramer`s rule. The rule naively implemented by Cramer is computationally inefficient for systems with more than two or three equations.  In the case of n equations in n unknowns, the calculation of n + 1 determinants is required, while the Gaussian elimination produces the result with the same computational complexity as the calculation of a single determinant.   [Verification required] Cramer`s rule can also be numerically unstable for 2×2 systems.  However, it has recently been shown that Cramer`s rule can be implemented with the same complexity as Gaussian elimination,  (systematically requires twice as many arithmetic operations and has the same numerical stability when the same permutation matrices are applied). In linear algebra, Cramer`s rule is an explicit formula for solving a system of linear equations with as many equations as unknowns, valid whenever the system has a single solution. It expresses the solution in terms of the determinants of the (quadratic) coefficient matrix and the matrices obtained from it by replacing a column with the column vector on the right side of the equations. It is named after Gabriel Cramer (1704–1752), who published the rule for a number of unknowns in 1750, although Colin Maclaurin also published special cases of the rule in 1748 (and perhaps knew this as early as 1729).

   The proof of Cramer`s rule uses the following properties of determinants: linearity with respect to a given column and the fact that the determinant is zero when two columns are equal, which is implied by the property that the sign of the determinant is reversed when you change two columns. An equation for ∂ x ∂ u {displaystyle {dfrac {partial x}{partial u}}} can be found by applying Cramer`s rule. However, this rule has some limitations in terms of solutions. This rule can only be applied if the system has unique solutions. But how do you know when a system has a unique solution? Let`s learn more about it, as well as the definition and formula of Cramer`s rule. The limits of Cramer`s rule are given below: This rule does not give the solution to the system of equations with infinite solutions and without solution. If D = 0, Cramer`s rule does not specify values for unknowns. It provides results only if D ≠ 0. The values of x, y, and z are calculated as follows.

Note that x is obtained by dividing the determinant of the matrix x by the determinant of the matrix of coefficients. This rule applies to the rest. When solving an AX = B system according to Cramer`s rule, if det A = 0, then the system has either an infinite number of solutions or no solution at all. In either case, we cannot close/find anything with Cramer`s rule. Indeed, when we find each variable with the formula of Cramer`s rule, we must divide the determinants by det A and a fraction is not defined if its denominator is 0. So if det A = 0, Cramer`s rule cannot be used. To solve a system of equations using Cramer`s rule, we first write it as AX = B. Then Dₓ is a determinant of Cramer`s rule of the coefficient matrix, where the first column is replaced by the column matrix B. In short, Cramer`s rule starts with a system of equations such as: Thus, Cramer`s rule helps us determine whether the given system has «no solution» or «an infinite number of solutions» by using the determinants we calculate to apply the rule. Go through the following example to learn how to solve Cramer`s rule for matrix 3×3.

No, Cramer`s rule doesn`t always work. As we know, it is only applicable if the given system of equations has a single solution. Cramer`s rule is one of the methods used to solve a system of equations. This rule contains determinants. that is, the values of the variables in the system are found using determinants. Consider a system of equations in n variables x₁, x₂, x₃, …, xn in the matrix form AX = B, where Cramer`s rule deals with determinants and determinants can only be found for quadratic matrices. But if we write 2×3 equations in the form AX = B, then A is NOT a quadratic matrix (it is a rectangular matrix) and therefore this rule cannot be applied in this case. Cramer`s rule can be used to prove that an integer programming problem whose stress matrix is completely unimodular and whose right side is integer has integer basis solutions. This makes the entire program much easier to solve. If the determinant of the matrix A in AX = B is 0, Cramer`s rule does not return the values of the variables.

Cramer`s rule states that the solution of the system of equations in the matrix form AX = B (where A is the matrix of coefficients, X is the column matrix of the variables and B is the column matrix of the coefficients) is obtained by dividing det (A) by the same determinant, where the respective columns are replaced by the matrix B. Cramer`s rule was invented by mathematician Gabriel Cramer in the 1750s. This rule is used to find the solution of a system of equations with any number of variables and the same number of equations. Sometimes, if we solve a system of equations in 3 variables, say x, y and z, we may need to solve x and y for two variables to solve for variable z. But with Cramer`s rule, we can find the value of any variable without finding the values of the other variables. Although Cramer`s rule does not help find the infinite number of solutions, we can determine whether the system has «no solution» or «an infinite number of solutions» by using the determinants we calculate as the process of applying the rule. Cramer`s rule applies when the determinant of the coefficient is non-zero. In case 2×2, if the coefficient determinant is zero, the system is incompatible if the counterdeterminants are non-zero, or indeterminate if the counterdeterminants are zero. Are you confused with this general formula of Cramer`s rule? Let`s look at this rule for 2×2 and 3×3 systems of equations for clarity.

Cramer`s rule is used to derive the general solution of a non-homogeneous linear differential equation by the variable parameter method. Cramer`s rule for matrix 2×2 is applied to solve the two-variable system of equations. Solve the following system of equations using Cramer`s rule: Suppose a1b2 − b1a2 is nonzero. Then, using the determinants x and y, Cramer`s rule can be used as From the table and explanation above, it is very clear that Cramer`s rule is NOT applicable if D = 0. that is, if the determinant of the coefficient matrix is 0, we cannot find the solution of the system of equations with Cramer`s rule. In this case, we can find the solution (if any) using the Gauss-Jordan method. We will now present a definitive method for solving systems of equations that uses determinants. This technique, known as Cramer`s rule, dates back to the mid-18th century. It is named after its innovator, the Swiss mathematician Gabriel Cramer (1704-1752), who introduced it in 1750.

Cramer`s rule is a viable and efficient way to find solutions for systems with any number of unknowns, provided we have the same number of equations as unknowns. Cramer`s rule is used in Ricci`s calculus in various calculations with Christoffel symbols of the first and second type.  A more general version of Cramer`s rule considers the matrix equation A brief proof of Cramer`s rule  can be given by stating that x 1 {displaystyle x_{1}} is the determinant of the matrix. Consider the map x = ( x 1 , . , x n ) ↦ 1 det A ( det ( A 1 ) , . , det ( A n ) ) , {displaystyle mathbf {x} =(x_{1},ldots ,x_{n})mapsto {frac {1}{det A}}left(det(A_{1}),ldots ,det(A_{n})right),} where A i {displaystyle A_{i}} is the matrix A {displaystyle A} with x {displaystyle mathbf {x} } in column i {displaystyle i}, as in Cramer`s rule. Because of the linearity of the determinant in each column, this map is linear. Note that it sends the column i {displaystyle i} from A {displaystyle A} to the base vector i {displaystyle i} ten e = ( 0 , . , 1 , . , 0 ) {displaystyle mathbf {e} _{i}=(0,ldots ,1,ldots ,0)} (with 1 at position i {displaystyle i}), since the determinant of a matrix with a repeating column is 0. So we have a linear map that coincides with the inverse of A {displaystyle A} on column space; therefore, it corresponds to A − 1 {displaystyle A^{-1}} over the column space range. Since A {displaystyle A} is invertible, the column vectors all include R n {displaystyle mathbb {R} ^{n}}, so our map is really the inverse of A {displaystyle A}.

Cramer`s rule follows. Cramer`s rule is one of the most important methods for solving a system of equations. In this method, the values of the variables in the system are calculated using the determinants of the matrices. Therefore, Cramer`s rule is also called the determining method. Let`s take a look at the formulas of Cramer`s rule for matrices 2×2 and 3×3. In particular, Cramer`s rule can be used to prove that the divergence operator on a Riemannian manifold is invariant with respect to the coordinate change. We give direct proof of this by removing the role of Christoffel`s symbols.

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