Let B 1 2 3 1 B is obtained when one applies the two elemen

Let B = [-1 2 3 1] B is obtained when one applies the two elementary row operations r^(3)_2 and then r^(-1)_21 successively to A. Find A A. [2 3 1 1/3] B. [0 7/3 1 1/3] C. [-2 1/3 1 1/3] D. [-4 1 1 1/3] E. none of these Which of the following statements about the following system of linear equations is true? x + y + z = 2 2x + y - z = 4 3x + 2y = a (i) it may have unique solution. (ii) it is consistent only when a is 5. iii) if it is consistent then there must be infinitely many solutions.

Solution

16. Let A =

a

b

c

d

After the elementary row operation r₂⁽³⁾ , A changes to

a

b

3c

3d

Also, after the elementary row operation r₂₁^(-1) ( interpreted as -1times 2^nd row added to the 1^st row) , the matrix A changes further to B=

a-3c

b-3d

3c

3d

Hence a-3c = -1, b-3d = 2, 3c = 3 and 3d = 1. Then a = 3c-1= 3-1= 2, b = 3d+2 = 1+2 = 3, c = 3c/3 = 3/3 = 1 and d = 3d/3 = 1/3.

Thus A =

2

3

1

1/3

Option A is the correct answer.

17. The augmented matrix of the given linear system is A =

1

1

1

2

2

1

-1

4

3

2

0

a

We will reduce A to its RREF as under:

Add -2 times the 1st row to the 2nd row

Add -3 times the 1st row to the 3rd row

Multiply the 2nd row by -1

Add-1 time the 2^nd row to the 1^st row

Add 1 times the 2nd row to the 3rd row

Then the RREF of A is

1

0

-2

2

0

1

3

0

0

0

0

a-6

Apparently, the given linear system will be consistent only if a = 6. Then it will have infinite solutions. The solutions will be x = 2+2z, y = -3z , z = z so that (x,y,z)^T = (2+2z, -3z,z)^T = (2,0,0)^T +z(2,-3,1)^T where z is an arbitrary real number. The 3^rd option is the correct answer.

a	b
c	d