Maths-

General

Easy

Question

# Suppose A and B are two nonsingular matrices such that

Hint:

### Linear equations are expressed using matrices, which are ordered rectangular arrays of numbers. Columns and rows make up a matrix. Mathematical operations like addition, subtraction, and matrix multiplication can also be done on matrices. We have given A and B are two nonsingular matrices and we have to find the correct condition as per the options.

## 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=BA^{2}

Lets simplify this we get:

B^{−1}(AB) = B^{−1}(BA^{2})

=(B^{−1}B)A^{2}^{2}

=A^{2}^{2}

A^{2}=B^{−1}AB

A^{4}=A^{2}A^{2}=(B^{−1}AB)(B^{−1}AB)=B^{−1}ABB^{−1}AB

=B^{−1}A(BB^{−1})A

=B^{−1}A(I)AB

=B^{−1}A^{2}B

=B^{−1}(B^{−1}AB)B

=B^{−2}AB^{2}

Now we have:

A^{8}=A^{4}A^{4}=(B^{−2}AB^{2})(B^{−2}AB^{2})

=B^{−2}A^{2}B^{2}^{2}=B^{−2}(B^{−1}AB)B^{2}

=B^{−3}AB^{3}

A^{16}=A^{8}A^{8}=B^{−3}B^{3}B^{−3}AB^{3}

=B^{−3}A^{2}B^{3}

=B^{−3}(B^{−1}AB)B^{3}

=B^{−4}AB^{4}

A^{32}=A^{16}A^{16}=B^{−4}AB^{4}B^{−4}AB^{4}

=B^{−4}A^{2}B^{4}

=B^{−4}(B^{−1}AB)B^{4}

=B^{−5}AB^{5}

as, B^{5}=I

so, B^{−5}=I

Now, putting in eqn=IAI

A^{32}=A

Here by multiplying by A^{−1} to both sides, we get:

A^{32}=A

A^{32}A^{−1}=A⋅A^{−1}

A^{31}=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 A^{31}=I.

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