How to deal with this problem of abstract matrix

First question:

Let A and B be n-order square matrices, and r(X) denote the rank of matrix X (r(X,Y) is the rank of the augmented matrix composed of X and Y), then which of the following relations is correct (the answer is A):

$$begin{eqnarray}
&(A)&; r(A, A B) = r(A) qquad
&(B)&; r(A, B A) = r(A)
&(C)&; r(A, B) = max {r(A), r(B)} qquad
&(D)&; r(A, B) = rleft(A^{T} B^{T}right)
end{eqnarray}$$

I use the following code to judge the a option, and I can’t get the correct result:

$Assumptions = Element[A | B, Matrices[{d, d}, Reals]]
MatrixRank[A] === MatrixRank[Join[A, A.B, 2]]

I wonder if there is any good way to solve this problem.

Second question:

It is known that the second-order matrix A has two different eigenvalues. $alpha_{1}$ and $alpha_{2}$ are two linearly independent eigenvectors of A and satisfy the condition of $A^{2}left(alpha_{1}+alpha_{2}right)=alpha_{1}+alpha_{2}$. Now I want to find the determinant value of matrix A (the answer is -1).

Assuming[Element[A, Matrices[{2, 2}, Reals]] && 
  Element[a1 | a2, Arrays[{2, 1}, Reals]], 
 Solve[A.a1 == k1.a1 && 
   A.a2 == k2.a2 && (A.A).(a1 + a2) == a1 + a2, {k1, k2}]]

But the above code cannot find the eigenvalues of matrix A, what should I do?