Obtain the eigenvalues and the corresponding eigenvectors of the matrix:
b) Diagonalized the matrix:
The eigenvalues of A are: λ1 = 2, λ2 =1 and λ3 = -1.
c) Show that gij is a contravariant tensor of rank 2.
d) Show that the vectors are linearly independent.
### (a) Eigenvalues और Eigenvectors निकालें: दिए गए मैट्रिक्स \( A = \begin{bmatrix}-2 & 5 & _________ _________ ______ ____ _____ ______.
___ ______ ____ _________ ________.
_____ _____ _______ ______ ________ _____ _____ ________ ____ ________ ____ __________.
________ ______ _________ ______ _____ ______ ____ _______ ___ ___ ____.
_____ _____ _________ ___ _______.
___ ______ ______ ___ ________ ___ ________ _______ _____.
______ ____ __________ _______ ____ _______ ________ ________ ___ ___.
__________ _______ ___ __________ ____ _________.
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Using separation of variables, solve:
where
Consider a rod of length L, whose ends are kept at a constant temperature and its lateral surface is insulated. The heat flow is described by the 1-D heat diffusion equation subject to the conditions:
Obtain a unique solution.
Obtain the Taylor series expansion of cos2 z about z = a.
Show that gij is a contravariant tensor of rank 2.
Show that the vectors are linearly independent.
Obtain the inverse Laplace transform of
Using appropriate figures, show that the group S3 is not commutative.