Let A = bmatrix 2 & 1 & 3 & 0 & 1 1 & 1 & 2 bmatrix , where > 0 . If P is a non-singular matrix of order 3 such that |P A^2 P⁻¹ - 2 P A P⁻¹ + I| = 144 , then the value of is

Step-by-step solution

The given expression inside the determinant is P A^2 P⁻¹ - 2 P A P⁻¹ + I . We can factor out P on the left and P⁻¹ on the right: P A^2 P⁻¹ - 2 P A P⁻¹ + I = P (A^2 - 2A + I) P⁻¹ = P (A - I)^2 P⁻¹ Taking the determinant of both sides: |P (A - I)^2 P⁻¹| = |P| |A - I|^2 |P⁻¹| = |A - I|^2 It is given that |A - I|^2 = 144 . Now, we find the matrix A - I : A - I = bmatrix 2 & 1 & 3 & 0 & 1 1 & 1 & 2 bmatrix - bmatrix 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 bmatrix = bmatrix 1 & 1 & 3 & -1 & 1 1 & 1 & 1 bmatrix Expanding the determ