Consider the following invertible matrix A and vector B A 6

Consider the following invertible matrix A, and vector B. A = [6 5 6 3 2 5 8 9 6 3 7 3 4 10 7 6 2 8 3 9 7 7 10 5 10], B = [3 6 10 3 5] Use Cramer\'s Rule to solve the following. Solve the equation AX = B for the variable x_4. Solve the equation AX = B for the variable x_3 .

Solution

For the given matrices A and B, we have det(A) = -5150

Also det(A₄) =   -5150 where A₄ is the matrix A with its 4^th column replaced by B ( as B is same as the 4^th column of A)

Further, det(A₃) = 0, where A₄ is the matrix A with its 3^rd column replaced by B ( as the 3^rd and the 4^th columns of A₄ are identical)

Hence, as per Cramer’s rule, x₄ = det(A₄)/det(A) = -5150/-5150 = 1. Also, x₃ = det(A₃)/det(A) = 0.