Determinants

Case I: If A is a nonsingular matrix, then its inverse exists. Now \(AX = B\) \(A^{-1}(AX) = A^{-1}B\) (premultiplying by \(A^{-1}\)) \((A^{-1}A)X = A^{-1}B\) (by associative property) \(IX = A^{-1}B\) \(X = A^{-1}B\) 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 \(|A| = 0\). In this case, we calculate \((adj A)B\). If \((adj A)B \neq O\), (O being zero matrix), then solution does not exist and the system of equations is called inconsistent. 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.