Question

If A and B are square matrices of same order and $A{A}^T=I$ then $(A^{T}BA{)}^{10}$ is equal to

• A. $A{B}^{10}{A}^T$
• B. $A^T{B}^{10}A$
• C. $A^{10}{B}^{10}(A^T{)}^{10}$
• D. $10A^{T}BA$

Answer 1

The correct option is B $A^T{B}^{10}A$

$A$ and $B$ are square matrices of same order and $A{A}^T=I$

let $x=A^{T}BA$

$x^2=A^{T}BA.A^{T}BA=A^T{B}^{2}A$

$x^3=A^T{B}^{2}A.A^{T}BA=A^T{B}^{3}A$

similarly,

$x^{10}=A^T{B}^{10}A$

Hence, option B.