Question

Determine if the given system is consistent. Do not completely solve the system.x1 + 2x3 =5x2 −3x4=3−2x2 + 2x3 +2x4 =42x1 + 9x4= −2Choose the correct answer below.
A. The system is consistent because the augmented matrix of the system can be reduced to triangular form.
B. The system is consistent because the augmented matrix of the system cannot be reduced to triangular form.
C. The system is inconsistent because the augmented matrix of the system can be reduced to triangular form.
D. The system is inconsistent because the augmented matrix of the system cannot be reduced to triangular form.

Answer 1

Answer:

$\left[\begin{array}{ccccc}1&0&2&0&:5\\0&1&0&-3&:3\\0&0&1&-2&:5\\0&0&0&\frac{-1}{4} &:-2\end{array}\right]$

Explanation

A. The system is consistent because the augmented matrix of the system can be reduced to triangular form.

Step-by-step explanation:

It can be reduced by triangular form using Row reduction method.

$\left[\begin{array}{ccccc}1&0&2&0&5\\0&1&0&-3&3\\0&-2&2&2&4\\2&0&0&9&-12\end{array}\right]$

The solution is in the attachment.

Answer 2

Question:

The given system of equations is

x1 + 2x3 + x4 = 5

x2 + x3 - 3x4 = 3

2x2 + 2x3 + 2x4 = 4

2x1 + 9x4= -2

Answer:

A. The system is consistent because the augmented matrix of the system can be reduced to triangular form.

Step-by-step explanation:

The system of equations can be solved in Matlab

The given 4x4 matrix is

A = [1 0 2 1; 0 1 1 -3; 0 -2 2 2; 2 0 0 9]

The given 4x1 matrix is

B = [5; 3; 4; -2]

We can check if the system can be reduced to lower and upper triangular form by using lu() function

[L,U] = lu(A)

Output:

L =

        0.5000         0              1.0000         1.0000

        0                  -0.5000    1.0000         0

        0                   1.0000     0                  0

        1.0000          0              0                  0

U =  

        2.0000         0              0               9.0000

        0                  -2.0000    2.0000    2.0000

        0                   0              2.0000    -2.0000

        0                   0              0              -1.5000

Therefore, the system is consistent because the given matrix of the system can be reduced to lower and upper triangular form.

We can find the corresponding solution of the system using lower and upper triangular matrices obtained in the previous step

x = U\(L\B)

Output:

x =

   2.0000

  -0.8333

   1.8333

  -0.6667

Alternatively:

The system can also be solved using linsolve() function

X = linsolve(A,B)

Output:

X =

      2.0000

    -0.8333

     1.8333

    -0.6667

As you can notice we got the same results as before.