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

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(credits to leowren)

step-by-step explanation:

Answer from: Quest

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