Question

CengageBrain - My Home MA 313, section 202. OL, Summer 2 2020 [0.27/0.83 Points] DETAILS PREVIOUS ANSWERS LARLINALG8 7.3.003. Prove that the symmetric matrix is diagonalizable. (Assume that a is real.) 00 a A= 0 a 0 а 0 0 Find the eigenvalues of A. (Enter your answers as a comma-separated list. Do not list the same eigenvalue multiple times.) X = 1,0,- 1 X Find an invertible matrix P such that p-1AP is diagonal. 0 -1 Ps 1 0 0 0 1 1 11 Which of the following statements is true? (Select all that apply.) A is diagonalizable because it is a square matrix. A is diagonalizable because it has a determinant of 0. A is diagonalizable because it is an anti-diagonal matrix. A is diagonalizable because it has 3 distinct eigenvalues. A is diagonalizable because it has a nonzero determinant.

Answer 1

equation Characteristic of A is o-X a-X O a 0 o-X {(x-x) (-x)} + a faca-x)} = 0 x (a-x) = a (a-x) = 0 (a-x) (x-a) - o (a-x) (+a) (x-a) = 0 - a, a Let Az =a to Let eigen vector corresponding value eigen x = – a AX - X, X - (A-X, I)x=0 a 6 0 Where A-X, I = 2a 0 a

We have solve it homogeneous system of equations and Gaussian Elimination. by O a 0 1 10 1 2x Ri O 0 0 ca O o 2a O - 0 Raxha Rz-aR, 0 O O a 0 a O O 00 x = -23 is Let vector 1:) &z=1, then eigen For vector da = a, let X [6] be eigen 6 o A X = 2X = (A - x2J) x = 0 A-X, I = where 0 o a o linear equation , have a homogeneous by Gaussian system of Elimination, he solve it O o O 1 0 o 0 1 O 6 o Rz-aR, 0 0 Sana 0 0 O 0 O O O 0 -a -a 6, -63 = 0 = x, =23 x = 23 x2 23 + O and Geometric Multiplicity Since Algebric Multiplicity is 1 of eigen value Geometric Multiplicity and of value -a is 2 Adgebric Multiplicity and eigen is diagonalizable So A

The matria with the eigen vectors coleemns is [ P - 0 1
A is diagonalisable because it is an anti-diagonal matrix.