A Note In this chapter, we restrict ourselves to the system of linear equations having unique solutions only. . . Solution of system of linear equations using inverse of a matrix Let us express the system of linear equations as matrix equations and solve them using inverse of the coefficient matrix.
Consider the system of equations a x + b y + c z = d a x + b y + c z = d a x + b y + c z = d Let A = , X and B Then, the system of equations can be written as, AX = B, i.e., Case I If A is a nonsingular matrix, then its inverse exists. Now AX = B A – (AX) = A – B (premultiplying by A – ) (A – A) X = A – B (by associative property) I X = A – B X = A – B This matrix equation provides unique solution for the given system of equations as inverse of a matrix is unique. This method of solving system of equations is known as Matrix Method. Case II If A is a singular matrix, then |A| = .
In this case, we calculate ( adj A) B. If ( adj A) B ≠ O, (O being zero matrix), then solution does not exist and the system of equations is called inconsistent. DETERMINANTS If ( adj A) B = O, then system may be either consistent or inconsistent according as the system have either infinitely many solutions or no solution. Example Solve the system of equations x + y = x + y = Solution The system of equations can be written in the form AX = B, where A = ,X and B Now, A = – ≠ , Hence, A is nonsingular matrix and so has a unique solution.
Note that A – = − Therefore X = A – B = – i.e.