Final Answer:
The conclusion from the above answer is that it is not possible to find a 2x2 matrix A and a 2-vector B such that the equation Ax = B has exactly two solutions. This is due to the Rank-Nullity Theorem, which states that the nullity (dimension of the null space) added to the rank of matrix A equals the number of columns in A. For a 2x2 matrix, the maximum nullity is 1, so it is impossible to have two solutions for this equation.
Step-by-step explanation:
No, it's not possible to have a 2-by-2 matrix A and a 2-vector B such that the equation Ax = B has exactly two solutions. The reason is related to the Rank-Nullity Theorem.
For any matrix A, the nullity (dimension of the null space) plus the rank of A is equal to the number of columns in A. In this case, A is a 2-by-2 matrix, so the number of columns is 2.
If Ax = B has exactly two solutions, it means that the nullity of A (dimension of the null space) is 2. However, the maximum possible nullity for a 2-by-2 matrix is 1, as the null space cannot have a dimension greater than the smaller of the two dimensions of the matrix.
Therefore, it is not possible to have a 2-by-2 matrix A and a 2-vector B such that Ax = B has exactly two solutions.