Question

3. (10 points) Determine the multiplicity of each eigenvalue and a basis and dimension of each eigenspace and state whether the matrix is diagonalizable or not. 1 6 -40 -7 0 0 -3 oni

Answer 1

The answer for above problem is explained below.

Given that, 4 16 Az -4 o -7 o -3 For Eigen values, 30 | A-XI 1 = - 4-1 6 2 ) - 4 -7 a o -3- (4-1) (3x+2841(12 +41) +6 (0) = 0 14-) » (3+) -4(3+1) = 0 (3+1) [42-1224] = 0 2) (2+3 ) (1-2)2=0 Eigen values are, Iz -3, 2, 2 The eigen values of A are, x =-3 c with algebraic multiplicity is 3) And 1202 (with algebraic multiplicity es 2 :)

The eigen vector corresponding to x, =-3 is, 4 + 3 6 le :: 31.6 O 6 T IT. -4 3 3:13 - 7 o matrix Augmented ės, 7 6 > -4 3 :: R₂ - 7 R₂ + 4R, 7 1 6 2 ) 25 -25 o 0 o R - R2 25 ㅖ 6 O -1 - :: o

: The - F eigen space, dim [E For d=2 Augmented matrix ės, 2 6 -4 -2 7 -5 R27 R2 + QR, 0 6 2 - = ) -5 B - R₂ + R₂ - ::: R, 1/2 Ri R- 1/2 R2

-ام 3 N o o o rector is Eigen polu 10 -- dim (E2] = 1 <2 Calgebraic multiplicity of 1 = 2) .. Matrix diagonalizable A is