Question 157757
Matrices are the most common and effective way to solve systems of linear equations.However, not all systems of linear equations have unique solutions.
First, it is important to establish whether a system in fact has a unique solution.
an example of a matrix that has no solution:

X+Y =2.........................1
2X+2Y=3.............................2

THIS HAS NO SOLUTION.....

Use row operations to show why it has no unique solution.
COEFFICIENT MATRIX AUGMENTED BY CONSTANTS IS 

1,1|2
2,2|3

ROW OPERATIONS

NR2=R2-R1*2

1,1,|2
0,0,|-1

RANK OF COEFFICIENT MATRIX = 1...........SINCE WE CAN GET A NON- ZERO DETERMINANT OF ORDER 1 ONLY FROM ROW REDUCED MATRIX.
NOT 2 THE ORDER OF THE MATRIX (2,2)

HENCE THERE ARE 2 POSSIBILITIES
1. NO SOLUTION.....AS IS HERE.

THIS HAPPENS WHEN RANK OF AUGMENTED MATRIX IS NOT EQUAL TO RANK OF COEFFICIENT MATRIX.
HERE RANK OF AUGMENTED MATRIX IS 2 SINCE A SECOND DEGREE NON ZERO 
DETERMINANT IS PRESENT ....VIZ
1,2
0,-1
SO WE SAY THE EQNS. ARE INCONSISTENT.