To understand why option 'B' is the correct answer, let's first define the terms involved:

- System of Equations: It is a collection of equations that are to be solved simultaneously. In this case, we have a system of n equations in n unknowns, represented as AX = B.

- Determinant (det): It is a scalar value that is computed from the elements of a square matrix. It provides important information about the properties of the matrix. In this case, we have the determinant of matrix A.

- Adjoint (adj): It is the transpose of the cofactor matrix of a given square matrix. In this case, we have the adjoint of matrix A.

- Zero Matrix (O): It is a matrix in which all the elements are zero.

Now, let's analyze each option one by one:

a) det.A ≠ 0:
If the determinant of matrix A is non-zero, it means that the matrix is invertible. In this case, the system AX = B will have a unique solution. Therefore, option 'a' is incorrect.

b) det.A = 0, (adj A) B = O:
If the determinant of matrix A is zero, it means that the matrix is singular or non-invertible. This implies that the system AX = B may have infinitely many solutions or no solution at all. To determine the number of solutions, we need to consider the product of the adjoint of A and B.

The adjoint of A, denoted as adjA, is a square matrix of the same size as A. The product of adjA and B, (adj A) B, is a matrix multiplication between adjA and B.

If (adj A) B is the zero matrix (O), it means that the system has infinitely many solutions. This is because the zero matrix represents a set of equations that are linearly dependent, allowing for multiple solutions. Therefore, option 'b' is correct.

c) det.A ≠ 0, (adj A) B ≠ O:
If the determinant of matrix A is non-zero and the product of adjA and B is non-zero, it implies that the system AX = B has a unique solution. Therefore, option 'c' is incorrect.

d) det.A = 0, (adj A) B ≠ O:
If the determinant of matrix A is zero and the product of adjA and B is non-zero, it implies that the system AX = B has no solution. Therefore, option 'd' is incorrect.

In conclusion, option 'B' is the correct answer because if the determinant of matrix A is zero and the product of the adjoint of A and B is the zero matrix, then the system AX = B will have infinitely many solutions.