please answer question 13 1415 16 with explaination Solve th

please answer question 13, 14,15 ,16 with explaination

Solve the system. 8x - 2y - 4z = -38 3x + 5y + 9z = 46 4x + 3y - 7z = -14 Solve the system. If there are infinitely many solutions, use x-arbitrary. -5x + 10y = 4 10x - 20y = -8 [3 0 -7 2 1 5 -4 6 9] Find -2 R_2 added to R_1 (show resulting matrix.) [1 0 4 8 -15 0 1 -6 7 9 0 0 1 6 -3 0 0 -3 4 -13] What row transformation is next? 3(R3) + R4 rightarrow R4 Solve the determinant. [5 2 3 3 4 0 1 -3 6]

Solution

12. The augmented matrix for the given linear system is A =

8

-2

-4

-38

3

5

9

46

4

3

-7

-14

To solve the given linear system, we will reduce A to its RREF as under:

Multiply the 1st row by 1/8

Add -3 times the 1st row to the 2nd row

Add -4 times the 1st row to the 3rd row

Multiply the 2nd row by 4/23

Add -4 times the 2nd row to the 3rd row

Multiply the 3rd row by -23/283

Add -42/23 times the 3rd row to the 2nd row

Add 1/2 times the 3rd row to the 1st row

Add 1/4 times the 2nd row to the 1st row

Then the RREF of A is

1

0

0

-2

0

1

0

5

0

0

1

3

Thus, the solution of the given linear system is x = -2,y = 5 and z = 3. We can substitute these values in any of the given equations to verify the result.

13. We have -5x+10y = 4…(1) and 10x -20y = -8…(2). On dividing both the sides of the 2nd equation by            -2, we get -5x +10y = 4. This is same as the 1^st equation. Thus, we have only one equation in 2 variables so that there will be infinite solutions. The solution is 10y = 4+5x or, y = 2/5 + x/2. Thus, the solution set is (x, /5 + x/2). The solutions are obtained by assigning arbitrary values to x.

14. The original matrix is A =

3

0

-7

2

1

5

-4

6

9

On adding -2 times the 2^nd row to the 1^st row(-2R₂ added to R₁), the matrix A changes to

-1

-2

-17

2

1

5

-4

6

9

15. The next row transformation is

Add 3 times the 3rd row to the 4th row

This will make the entry in the 4^th row,3^rd column 0. The matrix after this row operation changes to

1

0

4

8

-15

0

1

-6

7

9

0

0

1

6

-3

0

0

0

22

-22

16.Let the determinant’s matrix be denoted by A. Then det(A) = 5[4*6-0*(-3)]-2(3*6-0*1)+3[3*(-3)-4*1] = 5*24- 2*18+3*(-13) = 120-36-39 = 45

8	-2	-4	-38
3	5	9	46
4	3	-7	-14