Find the eigenvalues and eigenvectors of the matrix M , where M{2,2} = (1/2 2/3 ; 1/2 1/3) Hence express M in the form PDP^-1 where D is a diagonal matrix.

To start off, It's worth noting the definition of eigenvalues: for a Matrix A (n x n), it's ith eigenvalue (λ_i) is defined as the scalar constant the ith eigenvector (v_i) is multiplied for the matrix multiplication Av_i : Av_i = λ_iv (1)Hence, to find the ith eigenvalues, rearrange to get the equation: (A - λ_iI_n)v_i = 0 (2)Where I_n is the n x n Identity Matrix.For a non-trivial solution, A-λ_iI_i must be defined such that det(A-λ_iI_i) = 0Now we can solve the equation for λ:(1/2 - λ)(1/3 - λ) - (1/2)(2/3) = 0 (3)-The characteristic equation for A, with roots λ = 1, -1/6Now substitute each λ_i into equation (2) to solve for v_i where v_i = (x_i ; y_i).v_i should have no particular solution; there should be an infinite family of solutions. If this is not the case, an error has been made. Therefore, the eigenvector can take any value of x and y, with the ration of x : y constant. To solve, set the value of x_i (e.g. x_i = 1), then solve for y_i (or vice versa).Finally, the matrix D is the diagonal matrix of entries λ_i, and P the matrix of eigenvectors (be consistent with the order the ith eigenvalues and eigenvectors are entered):A = PDP^-1 (4)