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

1. is 4 and 2. is 3

step-by-step explanation:

so if you look at the first one you see that 1 is to 4 and 2 is to 8 and so on, so you can see for every 1 there is a times 4 so if it were 3 then it would be 3*4 and it is 12

for number 2 it is 3 because 1 is to 3 and 2 is to 6 so for every 1 there is 3 so if it were 3 then there would be 3*3 = 9

Answer from: Quest

pentagon

step-by-step explanation:

the coordinates a(5, -2), b(3,-1),c(-4,-4) d(-3,8) and e(-1,4) forms the vertices of the polygon when they are connected in order.

in order to classify that which polygon is it with the given coordinates, first count the number of vertices and then count the number of sides it can make with these vertices.

since, the given coordinates have five vertices, therefore, it will make five sides. moreover, pentagon is regular in nature therefore it will be convex.

now, a polygon with five sides is known as pentagon, therefore, the given polygon is the pentagon.

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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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