Lecture 39|| Review Ex2 Q9 Solve by Inversion Method ,Gauss Elimination ,Gauss Jordan ,Cramer’s Rule

(i) Matrix Inversion Method The Matrix Inversion Method is used to solve a system of linear equations in the form 𝐴 𝑋 = 𝐵 AX=B. If the coefficient matrix 𝐴 A is invertible, the solution is found by multiplying both sides B. This method works only when the determinant of 𝐴 A is non-zero. (ii) Gauss Elimination Method The Gauss Elimination Method solves a system of linear equations by transforming the augmented matrix into an upper triangular form using elementary row operations. Once in triangular form, back substitution is applied to find the values of the unknown variables. (iii) Gauss–Jordan Method The Gauss–Jordan Method is an extension of Gauss Elimination. It reduces the augmented matrix to reduced row echelon form (RREF) using row operations. This method directly provides the solution without the need for back substitution, as the coefficient matrix becomes the identity matrix. (iv) Cramer’s Rule Cramer’s Rule is a formula-based method to solve a system of linear equations using determinants. For 𝐴𝑋=𝐵 AX=B, the value of each variable is obtained by replacing the respective column of A B and dividing the determinant of this new matrix by det(A). This method applies only when A is a square matrix with a non-zero determinant.