Applications of Determinants and Matrices o Solving the system of linear equations o Checking the consistency of the system of linear equations .
Theorem : The determinant of the product of matrices is equal to product of their respective determinants , that is , | AB | = | A | * | B | , where A & B are square matrices of same order
Theorem : A square matrix A is invertible if and only if A is non singular matrix
Applications of Determinants and Matrices o Solving the system of linear equations o Checking the consistency of the system of linear equations .
Case1 : A is a non singular AX = B or X = A – 1 B
Case 2 : A is a singular matrix If A is a singular matrix , then | A | = 0 . AX = B or X = A – 1 B or X = ( 1 / | A |) * ( adj A ) * B In this case , we calculate ( adj A ) B . o If ( adj A ) B ≠ O , then solution does not exist and the system of equations is called inconsistent . o If ( adj A ) B = O , then system may be either consistent or inconsistent according as the system have either infinitely many solutions or no solution
Numerical : Examine Consistency x + 2y = 2 & 2x + 3y = 3 Solution :
| A | = -1 ≠ 0 , so A -1 exist . Thus it is consistent .