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.

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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

the most boring thing ever written

step-by-step explanation:

i am writing something. yes, i plan to make it the most boring thing ever written. i go to the store. a car is parked. many cars are parked or moving. some are blue. some are tan. they have windows. in the store, there are items for sale. these include such things as soap, detergent, magazines, and lettuce. you can enhance your life with these products. soap can be used for bathing, be it in a bathtub or in a shower. apply the soap to your body and rinse. detergent is used to wash clothes. place your dirty clothes into a washing machine and add some detergent as directed on the box. select the appropriate settings on your washing machine and you should be ready to begin. magazines are stapled reading material made with glossy paper, and they cover a wide variety of topics, ranging from news and politics to business and stock market information. some magazines are concerned with more recreational topics, like sports card collecting or different kinds of hairstyles. lettuce is a vegetable.

Answer from: Quest

Yes the slope to this is -5

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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 m...

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