- Books Name
- Mathmatics Book Based on NCERT
- Publication
- KRISHNA PUBLICATIONS
- Course
- CBSE Class 12
- Subject
- Mathmatics
Applications of Determinants and Matrices:
Consistency of System of Equations
Consistent system A system of equations is said to be consistent if its solution (one
or more) exists.
Inconsistent system A system of equations is said to be inconsistent if its solution
does not exist.
Solution of system of linear equations using inverse of a matrix
Suppose the system of equations is given by:
a1 x + b1 y + c1 z = d1
a2 x + b2 y + c2 z = d2
a3 x + b3 y + c3 z = d3
Now let us say, A, B and X are three matrices, such that;
or I X = A–1 B
or X = A–1 B
If A is a non-singular matrix, then X = A-1B.
This matrix equation provides unique solution for the given system of equations as inverse of a matrix is unique. This method of solving system of equations is known as Matrix Method.
Case-2 :
If A is a singular matrix, then determinant of A, |A| = 0.
Now for such a condition, there exist two cases based on (adj A) B.
- If (adj A) B O, (O being is zero matrix), then the system of equations does not have a solution and hence is called inconsistent.
- If (adj A) B = O, then the system of equations will have either consistent or inconsistent according as the system have either infinitely many solutions or no solution.
Problem :
Find if the given system of equations is consistent or inconsistent.
x+3y = 5 and 2x + 6y = 8
Solution: Given, the system of equations are:
x+3y = 5 and 2x + 6y = 8
As per the matrix Method, we know;
AX = B
Thus, (adj.A)B ≠ 0
Hence, the given system of equations is inconsistent.