Linear Algebra Answers

Using the the concept of rank of the matrix find whether the following system of equations

are consistent or inconsistent,

i.

4y + z = 0

12x - 5y - 3z = 34

-6x + 4z = 8

Using the the concept of rank of the matrix find whether the following system of equations

are consistent or inconsistent,

i.

4y + z = 0

12x - 5y - 3z = 34

-6x + 4z = 8

Consider the following augmented matrix [12]   1 −1 2 1 3 −1 5 −2 −4 2 x 2 − 8 x + 2   . Determine the values of x for which the system has (i) no solution, (ii) exactly one solution, (iii) infinitely many solutions.

If matrices

A =

1 5 3

2 5 7

B=

1

3

and C = [1 2].

compute A^tB, AC B^tA + CB, whenever defined. If you think any of these are not defined, give your reasons for saying so.

Consider the matrices A =   −2 7 1 3 4 1 8 1 5   ,B =   8 1 5 3 4 1 −2 7 1   , C =   −2 7 1 3 4 1 2 −7 3   . Find elementary matrices E1, E2 and E3 such that (5.1) E1A = B, (1) (5.2) E1B = A, (1) (5.3) E2A = C, (1) (5.4) E3C = A.

How do we show that W= {(x,-3x,2x)|x€R} is a subspace of R³?Also find a basis for subspace U of R³ which satisfies R³=W⊕U?

Evaluate T AB C− given that 2 2 0 1 2 3 3 1 , and 0 1 3 0 1 2 12 7 A B C       − − − = = =             

Given the system of equations 2 3 2 11 3 2 3 7 4 4 14 x y z x y z x y z + − = − − + = − + = , find the values of x, y, and z using matrix inversion. 

if M (x,y) = (x+y, x+y ) 

Find the matrix representation of M with respect to (v1,v2) where v1= (1,1) v2 = (1,3) 

prove that the vectors (1,0,0) ( 0,1,0) (1,1,0) is linearly dependent .