Final answer:
With a singular matrix A, we cannot conclude that A = B from the matrix equations A × F = B × F, A FB F, or FÃ = BF, because the multiplication by a singular matrix can lead to equal products even for different matrices A and B.
Step-by-step explanation:
The Impact of a Singular Matrix on Matrix Equations
When considering the question of whether matrix equations such as A × F = B × F lead us to conclude that A = B, it is critical to understand the properties of the matrices involved. In the case where A is a singular matrix, we encounter special circumstances that affect our reasoning. Here are the points that need to be discussed:
(a) If A × F = B × F, we cannot necessarily conclude that A = B, because if F is a zero matrix or another singular matrix that nullifies the columns, the products may be the same even for different A and B.
(b) Similarly, if A FB F, the same logic applies, and without additional information, A = B cannot be conclusively stated.
(c) In the equation FÃ = BF, we are faced with the transposed form of A (denoted Ã) being multiplied by F. Again, because multiplying by a singular matrix can yield a zero product, we cannot conclusively deduce that A = B from this equation alone.
The implications of a singular matrix in such matrix equations speak to broader algebraic principles, including the non-invertibility of singular matrices and their role in determining the solution sets of linear systems. Theorem FS and Theorem NME4 both relate to the behavior of non-singular (invertible) matrices and the unique solutions of linear systems, underscoring the fact that when dealing with singular matrices, such uniqueness may not hold.