Question

Show that if A and B are nxn matrices and N(A-B) is in Rⁿ, then A=B.

Answer 1

If the null vector is equal to the difference of two matrices A and B, then A is equal to B if and only if the corresponding components of vectors A and B are equal.

Step-by-step explanation:

Two vectors A and B are equal vectors if and only if their difference is the null vector:

Ổ = Ả – B = (AxÎ + AyĴ + AzÊ) – (Bxî + Byĵ + B₂k) = (Ax − Bx)Î + (Ay − By)Ĵ + (Az − Bz)Ê.

This vector equation means we must have simultaneously Ax - Bx = 0, Ay − By = 0, and Az − B₂ = 0.

Hence, we can write A = B if and only if the corresponding components of vectors A and B are equal: