Question

If A and B are two square matrices such that A²B = BA and (AB)¹⁰ = A^kB¹⁰. Then, k is ______.

Options

• 1001

• 1023

• 1042

• None of these

Answer 1

If A and B are two square matrices such that A²B = BA and (AB)¹⁰ = A^kB¹⁰. Then, k is 1023.

Explanation:

Here, (AB) (AB) = A(BA) B = A(A²B) B = A³B²

Now, (AB)(AB)(AB) = (A³B²)AB

= (A³B²AB) = A³B(BA)B

= A³B (A²B)B = A³(BA) . AB²

= A³(A²B) . AB = A⁵BAB² = A⁵ · A²B.B² = A⁷ . B³

So, (AB)ⁿ = `A^(2^n  –  1)` . Bn

∴ (AB)¹⁰ = `A^(2^10  –  1)` . B¹⁰

`\implies` k = 2¹⁰ – 1 = 1023