If A= [ array cc 1 & 2 - 2 & 1 array ] then A⁻¹=

Step-by-step solution

Matrix Inverse Computation The inverse of A = bmatrix a & b c & d bmatrix is given by A⁻¹ = 1 A adj (A) . For A = bmatrix 1 & 2 - 2 & 1 bmatrix , the determinant computes as A = 1 1 - ( 2 ) (- 2 ) = 1 + ^2 2 . Using the identity 1 + ^2 x = ^2 x , this simplifies to A = ^2 2 . The adjugate matrix is adj (A) = bmatrix 1 & - 2 2 & 1 bmatrix , which equals A^T . Combining these gives A⁻¹ = 1 ^2 2 A^T = ^2 2 A^T . Applying the half-angle identity ^2 x = 1 - 2x 2 yields ^2 2 = 1 - 2 , so A⁻¹ = 1 - 2 A^T . This matches op