Mathematics, 13.09.2019 18:30 cesar9895

A= ( −2 −1 2 −2 2 3 −4 1 3 ) b = ( −1 −1 4 ) x = ( x1 x2 x3 ) (a) (2 pts) write down the augmented matrix (a|b). (b) (4 pts) use gauss-jordan elimination to find the reduced row echelon form (rref) of the augmented matrix. (c) (2 pts) what is the rank of a? what is the rank of (a|b) (d) (2 pts) state whether the system is consistent or inconsistent. state how many solutions the system has; if there is/are a solution/s, write it/them down.

Answers: 2

Show answers

Answers

Answer from: kingjustin0825

The augmented matrix is $\left[\begin{array}{ccc|c}-2&-1&2&-1\\-2&2&3&-1\\-4&1&3&4\end{array}\right]$

The Reduced Row Echelon Form of the augmented matrix is $\left[\begin{array}{cccc}1&0&0&-3\\0&1&0&1\\0&0&1&-3\end{array}\right]$

The rank of matrix (A|B) is 3

The system is consistent and the solutions are $x_{1}= -3, x_{2} = 1, x_{3}= -3$

Step-by-step explanation:

We have the following information:

$A=\left[\begin{array}{ccc}-2&-1&2\\-2&2&3\\-4&1&3\end{array}\right], X=\left[\begin{array}{c}x_{1}&x_{2}&x_{3}\end{array}\right] and \:B=\left[\begin{array}{c}-1&-1&4\end{array}\right]$

1. The augmented matrix is

We take the matrix A and we add the matrix B we use a vertical line to separate the coefficient entries from the constants.

$\left[\begin{array}{ccc|c}-2&-1&2&-1\\-2&2&3&-1\\-4&1&3&4\end{array}\right]$

2. To transform the augmented matrix to the Reduced Row Echelon Form (RREF) you need to follow these steps:

Row operation 1: multiply the 1st row by -1/2

$\left[\begin{array}{cccc}1&1/2&-1&1/2\\-2&2&3&-1\\-4&1&3&4\end{array}\right]$

Row Operation 2: add 2 times the 1st row to the 2nd row

$\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&3&1&0\\-4&1&3&4\end{array}\right]$

Row Operation 3: add 4 times the 1st row to the 3rd row

$\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&3&1&0\\0&3&-1&6\end{array}\right]$

Row Operation 4: multiply the 2nd row by 1/3

$\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&1&1/3&0\\0&3&-1&6\end{array}\right]$

Row Operation 5: add -3 times the 2nd row to the 3rd row

$\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&1&1/3&0\\0&0&-2&6\end{array}\right]$

Row Operation 6: multiply the 3rd row by -1/2

$\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&1&1/3&0\\0&0&1&-3\end{array}\right]$

Row Operation 7: add -1/3 times the 3rd row to the 2nd row

$\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&1&0&1\\0&0&1&-3\end{array}\right]$

Row Operation 8: add 1 times the 3rd row to the 1st row

$\left[\begin{array}{cccc}1&1/2&0&-5/2\\0&1&0&1\\0&0&1&-3\end{array}\right]$

Row Operation 9: add -1/2 times the 2nd row to the 1st row

$\left[\begin{array}{cccc}1&0&0&-3\\0&1&0&1\\0&0&1&-3\end{array}\right]$

3. What is the rank of (A|B)

To find the rank of a matrix, we simply transform the matrix to its row echelon form and count the number of non-zero rows.

Because the row echelon form of the augmented matrix has three non-zero rows the rank of matrix (A|B) is 3

4. Solutions of the system

This definition is very important: "A system of linear equations is called inconsistent if it has no solutions. A system which has a solution is called consistent"

This system is consistent because from the row echelon form of the augmented matrix we find that the solutions are (the last column of a row echelon form matrix always give you the solution of the system)

$x_{1}= -3, x_{2} = 1, x_{3}= -3$

Answer from: Quest

i believe the correct answer is option 4.principalities were small territories that were run by russian rulers.

which statement correctly defines principalities in northern asia in the 1200s? principalities were small territories that were run by russian rulers.

hope i could ! : )

Answer from: Quest

overpriced

step-by-step explanation:

Another question on Mathematics

Mathematics, 21.06.2019 22:00

The evening newspaper is delivered at a random time between 5: 30 and 6: 30 pm. dinner is at a random time between 6: 00 and 7: 00 pm. each spinner is spun 30 times and the newspaper arrived 19 times before dinner. determine the experimental probability of the newspaper arriving before dinner.

Answers: 1

Answer

Mathematics, 22.06.2019 01:30

This graph shows a portion of an odd function. use the graph to complete the table of values. x f(x) −2 −3 −4 −6 im sorry i cant get the graph up

Answers: 1

Answer

Mathematics, 22.06.2019 02:30

What are the triple angle formulas? is it related to double angle identities?

Answers: 1

Answer

Mathematics, 22.06.2019 02:40

Which method of gathering data will be most effective in learning how diet and fitness choices affect long-term healt