Solving Linear Equations Using Matrix - Practice Questions & MCQ

Let us consider n linear equations in n unknowns, given as below

The above system of equations can be written in matrix form as 

Premultiplying equation AX=B by A^-1, we get

     A^-1(AX) = A^-1B ⇒ (A^-1A)X = A^-1B

                     ⇒ IX = A^-1B

                     ⇒  X = A^-1B 

                      ⇒   

Types of equation :

1. System of equations is non-homogenous:

   1. If |A| ≠ 0, then the system of equations is consistent and has a unique solution X = A^-1B

   2. If |A| = 0 and (adj A)·B ≠ 0, then the system of equations is inconsistent and has no solution.

   3. If |A| = 0 and (adj A)·B = 0, then the system of equations is consistent and has infinite number of solutions.

2. System of equations is homogenous:

   1. If |A| ≠ 0, then the system of equations has only one solution which is the trivial solution.

   2. If |A| = 0, then the system of equations has non-trivial solution and it has an infinite number of solutions.