Linear Algebra Answers

Inspect the following without finding the determinant:

1.

[1 0 0 0]

[0 1 0 0]

[0 0 1 0]

[0 0 0 - 2]

2. [1 0 0 0]

[0 1 0 0]

[0 0 0 1]

[0 0 1/4 0]

Show that if A is a matrix with a row of zeros,then A cannot have an inverse

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

ii.

5x - 3y + z = 7

2x + 3y - z = 0

8x + 9y - 3z = 2:

iii.

-8x + 2z = 1

6y + 4z = 3

12x + 2y = 2

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

ii.

5x - 3y + z = 7

2x + 3y - z = 0

8x + 9y - 3z = 2:

iii.

-8x + 2z = 1

6y + 4z = 3

12x + 2y = 2

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?