BACHELOR OF BUSINESS ADMINISTRATION

Detailed explanation-1: -Ax = b has a unique solution if and only if rank[A] = rank[A|b] = n. Calculation: Consider a non-homogeneous system of 5 linear equations in 4 variables, such that there is at least one variable with a non-zero coefficient in each equation.

Detailed explanation-2: -Solution of Non-homogeneous system of linear equations Matrix method: If AX = B, then X = A-1B gives a unique solution, provided A is non-singular. But if A is a singular matrix i.e., if |A| = 0, then the system of equation AX = B may be consistent with infinitely many solutions or it may be inconsistent.

Detailed explanation-3: -A non-homogeneous system of equations is a system in which the vector of constants on the right-hand side of the equals sign is non-zero. This lecture presents a general characterization of the solutions of a non-homogeneous system.

Detailed explanation-4: -A system of linear equations is said to be homogeneous if it can be written in the form Ax = 0, where A is an m × n matrix and 0 is the zero vector in Rm. Such a system always has at least one solution, namely x = 0 in Rn. This zero solution is usually called the trivial solution.

Detailed explanation-5: -Since, by the rank theorem, rank(A) + dim(N(A)) = n (recall that n is the number of columns of A), the system AX = B has a unique solution if and only if rank(A) = n. A linear system of the form AX = 0 is said to be homogeneous. Solutions of AX = 0 are vectors in the null space of A.