Question

Tutorial 6-Linear Systems EXERCISE .26. Solve the system x 3x1 +3x2, 32 2x1 + 4x2 subject to x1 (0) -, 2()5 by (1) diagonalisation of A (express the system as i - Ax), (2) using existence and uniqueness theorem and (3) calculating et in two ways.

Answer 1

Given,-3x + 3x, =2x +4% x (1)The system is in the matrix form is rx Where A and x = 2 4 Now consider the matrix A- the characteristic polynomial is det(A-11)-| now we need to find the eigen vectors for the corresponding eigen values det(4-(1)/)- for i -1 23人x2 so, the equation form is +-Xy = 0→ X,--X, So so, α- 2 | is a eigen vector for the corresponding eigenvalue 1 2 2x so, the equation form isx1-%, Ο is a eigenvector for the corresponding eigen value 6 so, a,

therefore the diagonal matrix is D- İfB-[4.4%, } is the basis set of V then T with respect to B is a diagonal matrix now the S matrix is: P2P and also PDPA 11 5 5 0 6 2 3 consider, PDP-1-2 2 4 Therefore, A is diagonalizable.