Given: AB=A and BA=B, where A and B are square matrices of the same order.
Multiply AB=A by A on the right:
ABA=A2
Rewrite the left side:
ABA=A(BA)
ABA=A⋅B
ABA=AB
ABA=A
Therefore:
A2=A
Since A2=A, all higher powers of A equal A:
A3=A2⋅A
A3=A⋅A
A3=A
Similarly, A4=A, A5=A, and so on.
For any positive integer n:
An=A
Therefore:
A2024=A
Multiply BA=B by B on the right:
BAB=B2
Rewrite the left side:
BAB=(BA)⋅B
BAB=B⋅B
BAB=B2
Also:
BAB=B(AB)
BAB=B⋅A
BAB=BA
BAB=B
Therefore:
B2=B
By the same reasoning as for A:
B2024=B
The value of A2024+B2024:
A2024+B2024=A+B
Therefore, the answer is A+B.