Linear Algebra Answers

a)Consider the funtion f:R\{−1}→R defined by f(x)=2x+1 x+1. i) ii) iii) iv) Check that f(x) is well defined and 1−1. Check that f(x)=2 for any x∈R. Check that g:R\{2}→R given by g(x)=x−1 Further,check that g(x)=−1 for any x∈R. 2−x is well defined and 1−1. (20) (3) (2) (4) Check that (f◦g)(x)=x for x∈R\{2} and (g◦f)(x)=x for x∈R\{−1}.(4) b)Find the direction cosines of the perpendicular from the origin to the plane r·(6i+4j+2√ 3k)+2=0.

a)Show that, if A is any n×n matrix with real entries, then there is a n×n symmetric matrix S and a n×n skew symmetric matrix S' such that A=S+S'.(3)b)Find the solutions to the following system of equations by reducing the corresponding augmented matrix to row-reduced echelon form.
2a+3b+4c+d=8
a+2b+2c+2d=3
a−b+c+3d=3

a)Find the values of a,b∈C for which the matrix (1. i. 1+i)
( a. 0. b. )
(1−i. 2+i. 1) is Hermitian.(2)b)
Are there values of a∈C for which the matrix (1. 0. 0 )
(0. −1/√2. 1/√2)
(0. 1/√2. a. )
is unitary? Justify your answer.(3)c)Let (x1,x2,x3) and (y1,y2,y3) represent the coordinates with respect to the bases B1={(1,0,0),(0,1,0),(0,0,1)},B2={(1,0,0),(0,1,2),(0,2,1)}.If Q(X)=x21+2x1x2+2x2x3+x22+x23,find the representation of Q in terms of (y1,y2,y3).

For the following matrices, check whether the reexists an invertible matrix P such that P−1 A Pis diagonal. When such a P exists,find P.
i)A=( 0 1 -3)
( 2 -1 6)
( 1 -1 4)
(ii) B is equal to. ( 0 0 -2)
( 1 2 1)
( 1 0 3)

b)Find the inverse of the matrix B in part (a) by using Cayley-Hamilton theorem.(3)c)Using the fact that det(AB)=det(A)det(B)for any two matrices A and B,prove the identity (a2+b2) (c2+d2)=(ac−bd)2+(ad+bc)2

4)a) Show that, if A is any n×n matrix with real entries, then there is a n×n symmetric matrix S and a n×n skew symmetric matrix S' such that A=S+S'.
b)Find the solutions to the following system of equations by reducing the corresponding augmented matrix to row-reduced echelon form.
2a+3b+4c+d=8
a+2b+2c+2d=3
a−b+c+3d=3

) Consider the funсtion f:R\{−1}→R defined by f(x)=2x+1 /x+1.
i) Check that f(x) is well defined and 1−1.
2)Check that f(x) is not =2 for any x∈R.
3)Check that g:R\{2}→R given by g(x)=x−1/2-x is well defined and 1-1 Further,check that g(x)=−1for any x∈R.
(4) Check that (f◦g)(x)=x for x∈R\{2} and (g◦f)(x)=x for x∈R\{−1}.
b)Find the direction cosines of the perpendicular from the origin to the plane r·(6i+4j+2√ 3k)+2=0.

Integrate with respect to x :
∫10xdx

a 4

b 1

c 2

d 1/2

show the matrix [ r1= -9 4 4 ,r2= -8 3 4,r3= -16 8 7,] of order 3*3 is diagonalizable.Obtain the diagonalizable matrix P.

What is the difference between Row Echelon form and Reduced Row Echelon form? Also, what is the difference between Gauss Elimination and Gauss Jordon elimination method?

a floor manager is going to install two types of machine, small and large. the following table shows the number of operators and the space requirements for each machine:

Small Large Maximum available
Number of operators 5 4 40
Space in m2 2 4 28

i) Taking x to represent the number of small machines and y to represent the number of large machines, write down two inequalities in x and y and illustrate these on a graph.

ii) If the profit on each small machine is €120 per day and the profit on each large machine is €200 per day, calculate the number of each type of machine that should be installed in order to have a maximum profit. What is the profit?