Suppose A and B are two nonsingular matrices such that

The correct answer is:

A non-singular matrix is a square matrix with a non-zero value for the determinant. To determine a matrix's inverse, the non-singular matrix property must be met.
Now we have given 
AB=BA²
Lets simplify this we get:
B⁻¹(AB) = B⁻¹(BA²)
=(B⁻¹B)A²²
=A²²
A²=B⁻¹AB
A⁴=A²A²=(B⁻¹AB)(B⁻¹AB)=B⁻¹ABB⁻¹AB
=B⁻¹A(BB⁻¹)A
=B⁻¹A(I)AB
=B⁻¹A²B
=B⁻¹(B⁻¹AB)B
=B⁻²AB²
Now we have:
A⁸=A⁴A⁴=(B⁻²AB²)(B⁻²AB²)
=B⁻²A²B²²=B⁻²(B⁻¹AB)B²
=B⁻³AB³
A¹⁶=A⁸A⁸=B⁻³B³B⁻³AB³
=B⁻³A²B³
=B⁻³(B⁻¹AB)B³
=B⁻⁴AB⁴
A³²=A¹⁶A¹⁶=B⁻⁴AB⁴B⁻⁴AB⁴
=B⁻⁴A²B⁴
=B⁻⁴(B⁻¹AB)B⁴
=B⁻⁵AB⁵
as, B⁵=I
so, B⁻⁵=I
Now, putting in eqn=IAI
A³²=A

Here by multiplying by A⁻¹ to both sides, we get:
A³²=A
A³²A⁻¹=A⋅A⁻¹
A³¹=I

Here we used the concept of matrices to understand the problem and the concept. The different types of matrices are Square matrix, Diagonal matrix, Zero matrix, Symmetric matrix, Identity matrix, Upper triangular matrix, Lower Triangular Matrix. Here the correct option is A³¹=I.