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Problem 17: State the conditions under which a system of non-homogeneous equations will have i. no solution ii. a unique solution iii. infinity of solutions

Problem 17: State the conditions under which a system of non-homogeneous equations will have i. no solution ii. a unique solution iii. infinity of solutions.

Solution: 

Let $AX=B$ be a system of linear non-homogeneous equations, where $A,X,B$ are $m\times n,n\times 1,m\times 1$ matrices respectively. 

 i. These equations will have no solution if the coefficient matrix $A$ and augmented matrix $\left[A|B\right]$ are not of the same rank. 

 ii. These equations will possess a unique solution if the matrices $A$ and augmented matrix $\left[A|B\right]$ are of the same rank and the rank is equal to the number of variables. 

 iii. These equations will have infinity of solutions if the matrices $A$ and augmented matrix $\left[A|B\right]$ are of the same rank and the rank is less than the number of variables.

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