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.