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Question 157757This question is from textbook Precalculus
: Provide an example of a matrix that has no solution. Use row operations to show why it has no unique solution. Also, some matrices have more than one solution (in fact, an infinite number of solutions) because the system is undetermined. (In other words, there are not enough constraints to provide a unique solution.) Provide an example of such a matrix, and show, using row operations, why it is underdetermined.
This question is from textbook Precalculus
Answer by MathLover1(20850)   (Show Source):
You can put this solution on YOUR website! 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.
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