Chapter 10, Problem 16RCC

To determine

To state: Cramer’s Rule for solving a system of linear equation.

Answer to Problem 16RCC

The solution is given by $x_i=\frac{\det \left(A_{x_i}\right)}{\det \left(A\right)}$ .

Explanation of Solution

Given:

Calculation:

Cramer's Rule states the following:

If A is a $n\times n$ square matrix such that $\det \left(A\right)\ne 0$ , then the linear system of equations represented by $AX=B$ can be solved using determinants.

That is, for each variable $x_i$ appearing in the equation, the solution is given by

  $x_i=\frac{\det \left(A_{x_i}\right)}{\det \left(A\right)}$

Where the matrix $A_{x_i}$ is obtained by replacing the ith column of A with B.

Cramer's Rule is useful when the matrix A from the equation $AX=B$ is a square matrix as it simplifies the issue of computing the denominators of the scalars that appear in Gaussian elimination.

However, Cramer's Rule becomes tedious for larger systems as compute the determinant of large matrices involves a lot of expansions along cofactors.

In addition, Cramer's Rule is limited to square matrices and so it provides us with no information regarding other matrices. Gaussian elimination, on the other, is applicable to any matrix regardless of whether a solution is exists or is unique.

Conclusion:

Therefore, the solution is given by $x_i=\frac{\det \left(A_{x_i}\right)}{\det \left(A\right)}$ .