Question

Then diago- 6. Find the eigenvalues and eigenvectors of the matrix A = nalize the matrix. [4 points)

Answer 1

Ane) A = [1 s find eign value The cign value are those a for which IA-XIO *** det (a-at) - det([15] -66:]) det([*?3-0 : so - (1-4) E-2-2) + 5 = (2-4) (2+2)+5 = A2-21-8+5 = A2 -2 2-3 (A-I = 0 22 - 2a-3=0 12- zata-3=0 (2-3) (2+1) = 0 2-3,-1 Bo the cegn value of Matrix A = 3,- 3-1 And

Find eign Vector if IA-al=0 then the equation b=0 (A-AI) x = b AX AX for A=3 the oan AX=ax becomes AX=3x writing x=(x 7 AX= [-2 -5 7x7 t X2 - [-22 X -542 + 4X2 7 while x = Bx, 7 1 342 Selting those equal , we get - 2x - 5X2 = 3x = 5*,= -5 X2 33 x, = -X2 X+ 4X2 = 3x2 = x1 = - X2 this means that, while are infinitely many non zero solution (solution vector of the equation AX=3x, they all satisfy the condition that the first entry x, és -1 teines of second entry X2 . this all solution of this ean can be characterized by

entrere t es any voal No..the nonzero vectory that Sabesfy AX = -3x que called eign vector associated with the eign value a=3, one such eignvectoras ford=-1 the con AX Ax = = -X x becomes AX = [-2x, -5x2) IX, +4x2 while *= ER selling shose equal -2x, 5x27-ax7. L****e] E] - 2X, - 5x2 = -x, = -x, = 5x2 3x = -5x2 xet 4x2 = -X2 = x, = -5x2 Hence the set of eignvector associated with a=1 es spamed by uy = Es Day

Diagonalize the matrix A=[-2-5 by PDP-1=A the eignvalue of A are 34-1. correspond eign vector are alle Esmosbectivity, P-E-5] =1P1= -4 pa 20. P-7538JE So. ] = Aug