Question

I need help with two of these problems

Find a matrix P that orthogonally diagonalizes A = -1 201 2 02, then find PT AP. 0 2 1]

Answer 1

2nd Giren that a mys A= 3 det (A) it A is orthogonal -0. Then , CO. NTO det(A) = a. the con la ate 9 wro Ando aa no 1st ! dod 2 Given that, Todo oda A= -- we have to find P soti PAPT is diagonalo, Now let us find the eigen values of A. ( A – A1) = 0 , det. -1-1 2 2 2 2 5 E1-03 4-449-47-272-29120 A-ty of Ah - 1² +41 - 4+40=e, ax3 +94 ao a cing) = =0,3, -3,

rectors, Now gen we have to find the corresponding ei for a=0 (A-T)* -0. 17 do a 2 -2 Ng 0 x + ad -zy 2X + 22 EO. + 2 = 0 zy C net y = I tre 2 =-20 2 !) Vi tyg C - -2 for a=3 (A-AI)* 2 -4 2. 2 -2 2 2 = 40 +29 21 240 -3y + 28 = ay Let y=c 2-C -27 = o. x 2 3 ya X- C [1] y Vy =

for 93-3. A-AI) x = 0. 2 2 76-6 2 2 4 3 o 2 22 t try By + 27 24 t 2y + 4z=e. z = old y=-e ket x=C 1 X= C WS an "2 Hence, p= [" 2 r3 L 1 a dad INT -1 ya Ans