If A= [ array cc 1 & x - x & 1 array ] , then A^T A⁻¹=

Step-by-step solution

aligned & aligned |A| & = | array cc 1 & x - x & 1 array | & =1+ ^2 x 0 aligned & If A= [ array ll a & b c & d array ] and ad -b c 0, & then A⁻¹= 1 (a d-b c) [ array cc d & -b -c & a array ] aligned aligned & A ⁻¹= 1 1+ ^2 x [ array cc 1 & - x x & 1 array ] & A ^ T A ⁻¹= [ array cc 1 & - x x & 1 array ] [ array cc 1 1+ ^2 x & - x 1+ ^2 x x 1+ ^2 x & 1 1+ ^2 x array ] aligned aligned & = [ array ll 1- ^2 x 1+ ^2 x & -2 x 1+ ^2 x 2 x 1+ ^2 x & 1- ^2 x 1+ ^2 x array ] & = [ array ll 2 x & - 2 x 2 x & 2 x array ] align