Let M = pmatrix ^2( ⁻¹ ) & 2 -2 & cosec ^2( ⁻¹ ) pmatrix be a 2 2 matrix, where > > 0 . If the trace of the matrix M is 22 and the determinant of M is 89 , then the value of ^2 + ^3 is equal to

Step-by-step solution

Using the identities ^2( ⁻¹ ) = 1 + ^2 and cosec ^2( ⁻¹ ) = 1 + ^2 , the matrix M can be written as: M = pmatrix 1 + ^2 & 2 -2 & 1 + ^2 pmatrix The trace of matrix M is the sum of its diagonal elements: Trace (M) = (1 + ^2) + (1 + ^2) = 22 ^2 + ^2 = 20 The determinant of matrix M is: Det (M) = (1 + ^2)(1 + ^2) - (2)(-2) = 89 (1 + ^2)(1 + ^2) + 4 = 89 (1 + ^2)(1 + ^2) = 85 Let X = 1 + ^2 and Y = 1 + ^2 . We have the system of equations: X + Y = 22 XY = 85 Thus, X and Y are the roots of the quadratic equation t^2 - 2