5. Applications of Determinants and Matrices

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.