Write the system as an augmented matrix Use reduced row eche

Write the system as an augmented matrix. Use reduced row echelon to find the solution of the linear system. -xi + 3x_3 + x_4 = 2 2x_1 + 3x_2 - 3x_3 + x_4 = 2 2x_1 - 2x_2 - 2x_3 - x_4 = -2 Determine all values of b such that [1 2 a 0][3 b -4 1] = [-5 6 12 16]

Solution

4. The given linear system of equations can be represented in nthe matrix form as AX = b, where b =            (2,2,-2)^T , X = (x₁,x₂,x₃,x₄)^T and the coefficient matrix is A =

-1

0

3

1

2

3

-3

1

2

-2

-2

-1

The augmented matrix of the given linear system of equations is B (say) = [A,b] =

-1

0

3

1

2

2

3

-3

1

2

2

-2

-2

-1

-2

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

Multiply the 1st row by -1

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

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

Multiply the 2nd row by 1/3

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

Multiply the 3rd row by 1/6

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

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

Then the RREF of B is

1

0

0

½

1

0

1

0

½

1

0

0

1

½

1

Then the given linear system of equations is equivalent to x₁+x₄/2=1, or, x₁=-x₄/2, x₂+x₄/2=1 or, x₂=-x₄/2               and x₃+x₄/2=1 or, x₃=-x₄/2.Then X =(-x₄/2, -x₄/2,-x₄/2,x₄)^T = x₄/2(-1,-1,-1,2)^T or, X = t(-1,-1,-1,2)^T, where t = x₄ is an arbitrart real number. Thus, the given linear system of equations has infinite solutions of the form t(-1,-1,-1,2)^T.

5. On multiplication of the 2 matrices on the left hand side, we get the matrix

-5

b+2

3a

ab

Therefore,b+2 = 6 or, b = 4, and 3a = 12 or, a = 4. Thus, b = 4.

-1	0	3	1
2	3	-3	1
2	-2	-2	-1