Final answer:
The question involves determining whether equal matrix products imply matrix equality in linear algebra. It depends on the properties of the matrices involved, specifically the invertibility of the matrix F. Simply having equal products does not guarantee matrix equality without further conditions.
Step-by-step explanation:
The question refers to matrices and solution sets in the context of linear algebra, a branch of mathematics. It deals with the concept of whether equal products involving matrices imply the matrices themselves are equal. The notations A, F, and B appear to represent matrices, while x represents a column vector, and b_f implies a particular vector in the field denoted by F^m. The typos or irrelevancies noted in the equation examples such as In A and F~ F(n₁ – 1, n₂ − 1), should be ignored to focus on the main topic which is about matrix equality and implications of product operations.
To address the questions (a), (b), and (c) we must employ the properties of matrix operations. In part (a), we cannot conclude A = B merely because A × F = B × F unless F is an invertible matrix. For part (b), the expression is unclear due to a typo but typically, if the matrix F is non-singular and A FB F, this does not directly imply A = B unless other conditions are met. Lastly, in part (c), FÃ = BF also does not guarantee A = B unless F is invertible.
The conclusion about matrix equality depends on additional conditions such as the invertibility of the matrix F. Without such conditions, we cannot conclude matrix equality solely based on equal products.