If A = bmatrix 2i & i^3 i^2 & 1 bmatrix , then A⁻¹ is equal to

Step-by-step solution

Given A = bmatrix 2i & i^3 i^2 & 1 bmatrix Substituting i^2 = -1 and i^3 = -i , we get: A = bmatrix 2i & -i -1 & 1 bmatrix The determinant of A is: |A| = (2i)(1) - (-i)(-1) = 2i - i = i The adjugate of A is: adj (A) = bmatrix 1 & i 1 & 2i bmatrix The inverse of A is given by A⁻¹ = 1 |A| adj (A) A⁻¹ = 1 i bmatrix 1 & i 1 & 2i bmatrix Since 1 i = -i , we have: A⁻¹ = -i bmatrix 1 & i 1 & 2i bmatrix = bmatrix -i & -i^2 -i & -2i^2 bmatrix Substituting i^2 = -1 : A⁻¹ = bmatrix -i & 1 -i & 2 bmatrix Answer: bmatrix -i & 1