Let A = bmatrix a & 1 1 & b bmatrix , where a and b are the roots of the equation x^2 - 4x + 2 = 0 . If A + A⁻¹ = kI₂ , then the value of k is ____

Step-by-step solution

Given A = bmatrix a & 1 1 & b bmatrix Since a and b are roots of x^2 - 4x + 2 = 0 , we have a + b = 4 and ab = 2 . The determinant of A is |A| = ab - 1 = 2 - 1 = 1 . The inverse of A is A⁻¹ = 1 |A| bmatrix b & -1 -1 & a bmatrix = bmatrix b & -1 -1 & a bmatrix . Now, A + A⁻¹ = bmatrix a & 1 1 & b bmatrix + bmatrix b & -1 -1 & a bmatrix = bmatrix a+b & 0 0 & a+b bmatrix . Substituting a + b = 4 , we get A + A⁻¹ = bmatrix 4 & 0 0 & 4 bmatrix = 4I₂ . Comparing with A + A⁻¹ = kI₂ , we get k = 4 .