6. For matrix $A=⎣⎡ 20−2 −14−2 −101 ⎦⎤ $. a. Show that $⎣⎡ 10−1 ⎦⎤ $ is an eigenvector of $A$ and find the corresponding eigenvalue. (4) b. Show that 4 is an eigenvalue of $A$ and find an eigenvector associated with it. (4) c. Show that $def(A)=0$. (4) d. Is $A$ invertible? Briefly explain! e. From the observation that $der(A)=0$, deduce the value of the third eigenvalue of $A$ and find an eigenvector associated with it.

(a)Characteristic equation

(2- ){(4- (1-

Thus, the EIGEN VALUE OF MATRIX A are 0,3,4

when the corresponding eigen vector is given by

-x-y-z=0 .......(1)

0.x+y+0.z=0 ..........(2)

-2x-2y-2z=0 ......(3)

from eqn.(2)

y=0

put y=0 in eqn.(3)

then we get -x-z=0

-x=z .....(4)

therefore put x=1 in eqn.(4) we get z=-1

then the eigen vector= hence proved.

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