7.0 Types of Equations Homogenous & Non-Homogenous

  Matrices and Determinants

• If $|A| \ne 0$,then the system of equations is consistent and has a unique solution given by $X = {A^{ - 1}}B$.
• If $\left| A \right| = 0$ and $(adjA).B \ne 0$,then the system of equations is inconsistent and has no solution.
• If $\left| A \right| = 0$ and $\left( {adjA} \right).B = 0$ then system may be either consistent or inconsistent according as the system have either infinitely many solutions or no solution.

• If $|A| \ne 0$,the system of equations have only trivial solution and it has one solution.
• If $|A|=0$,the system of equations has non trivial solution and it has infinite solutions.
• If number of equations < number of unknowns, then it has non trivial solution.