Let A=⎡⎢⎣2070101−21⎤⎥⎦ and B=⎡⎢⎣−x14x7x010x−4x−2x⎤⎥⎦ be two matrices such that AB=(AB)−1 and AB≠I, where I is an identity matrix of order 3×3. Then the value of tr(AB+(AB)2+(AB)3+⋯+(AB)100) is( Here, tr(A) denotes the trace of matrix A, i.e., sum of diagonal elements of A.)

Let A=⎡⎢⎣2070101−21⎤⎥⎦ and B=⎡⎢⎣−x14x7x010x−4x

1.	Let A=⎡⎢⎣2070101−21⎤⎥⎦ and B=⎡⎢⎣−x14x7x010x−4x−2x⎤⎥⎦ be two matrices such that AB=(AB)−1 and AB≠I, where I is an identity matrix of order 3×3. Then the value of tr(AB+(AB)2+(AB)3+⋯+(AB)100) is( Here, tr(A) denotes the trace of matrix A, i.e., sum of diagonal elements of A.)
Answer» Let $A = ⎡ ⎢ ⎣ \begin{matrix} 2 & 0 & 7 0 & 1 & 0 1 & - 2 & 1 \end{matrix} ⎤ ⎥ ⎦$ and $B = ⎡ ⎢ ⎣ \begin{matrix} - x & 14 x & 7 x 0 & 1 & 0 x & - 4 x & - 2 x \end{matrix} ⎤ ⎥ ⎦$ be two matrices such that $A B = (A B)^{- 1}$ and $A B \neq I,$ where $I$ is an identity matrix of order $3 \times 3.$ Then the value of $t r (A B + (A B)^{2} + (A B)^{3} + \dots + (A B)^{100})$ is $($ Here, $t r (A)$ denotes the trace of matrix $A,$ i.e., sum of diagonal elements of $A .)$