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Solve the following
0.6x + 0.8y + 0.1z = 1
1.1 x + 0.4y + 0.3z= 0.2
x + y + 2z =0.5
by LU decomposition method and find the inverse of the coefficient matrix
Find the orthogonal canonical reduction of the quadratic form
x
2 +y
2 +z
2 −2xy−2xz−2yz. Also, find its principal axes.
Check whether the matrices A and B are diagonalisable. Diagonalise those matrices
which are diagonalisable.
i) A =


−2 −5 −1
3 6 1
−2 −3 1

 ii) B =


−1 −3 0
2 4 0
−1 −1 2

.
b) Find inverse of the matrix B in part a) of the question by finding the adjoint as well
as using Cayley-Hamiltion theorem.
 ii) B =


−1 −3 0
2 4 0
−1 −1 2

.
b) Find inverse of the matrix B in part a) of the question by finding the adjoint as well
as using Cayley-Hamiltion
Consider the basis e1 = (−2,4,−1), e2 = (−1,3,−1) and e3 = (1,−2,1) of R
3
over R. Find the dual basis of {e1, e2, e3}.
Check whether the vector (2root3, 2 ) is equally inclined to the vectors (2, 2root3) and (4,0)
Check whether the vector (2root3, 2 ) is equally inclined to the vectors (2, 2root3) and (4,0)
Check whether the vector (2root3, 2 ) is equally inclined to the vectors (2, 2root3) and (4,0)
which of the following statements are true and which are false ? justify your answer with a short proof or a counterexample. i) r^2 has infinitely many non zero, proper vector subspaces. ii) if t:v -> w is one-one linear transformation between two finite dimensional vector spaces v and w then t is invertible. iii) if a^k = 0 for a square matrix a, then all the eigen values of a are non zero. iv) every unitary operator is invertible. v) every system of homogeneous linear equations has a non zero solution.
define t : r^3>r^3 by t(x,y,x)=(x+y,y,2x-2y+2z) check that t satisfies the polynomial (x-1)^2(x-2). find the minimal polynomial of t.
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