If A = bmatrix 1 & - 2 2 & 1 bmatrix and B = bmatrix 1 & 2 - 2 & 1 bmatrix then A⁻¹B is equal to

Step-by-step solution

Let t = 2 . The given matrices can be written as: A = bmatrix 1 & -t t & 1 bmatrix and B = bmatrix 1 & t -t & 1 bmatrix The determinant of A is |A| = 1(1) - (-t)(t) = 1 + t^2 . The adjugate of A is adj (A) = bmatrix 1 & t -t & 1 bmatrix . Therefore, the inverse of A is: A⁻¹ = 1 |A| adj (A) = 1 1+t^2 bmatrix 1 & t -t & 1 bmatrix Now, multiply A⁻¹ and B : A⁻¹B = 1 1+t^2 bmatrix 1 & t -t & 1 bmatrix bmatrix 1 & t -t & 1 bmatrix A⁻¹B = 1 1+t^2 bmatrix 1(1) + t(-t) & 1(t) + t(1) -t(1) + 1(-t) & -t(t) + 1(1) bmatrix A⁻¹B