Respuesta :
Answer:
The augmented matrix is [tex]\left[\begin{array}{ccc|c}-2&-1&2&-1\\-2&2&3&-1\\-4&1&3&4\end{array}\right][/tex]
The Reduced Row Echelon Form of the augmented matrix is [tex]\left[\begin{array}{cccc}1&0&0&-3\\0&1&0&1\\0&0&1&-3\end{array}\right][/tex]
The rank of matrix (A|B) is 3
The system is consistent and the solutions are [tex]x_{1}= -3, x_{2} = 1, x_{3}= -3[/tex]
Step-by-step explanation:
We have the following information:
[tex]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][/tex]
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.
[tex]\left[\begin{array}{ccc|c}-2&-1&2&-1\\-2&2&3&-1\\-4&1&3&4\end{array}\right][/tex]
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
[tex]\left[\begin{array}{cccc}1&1/2&-1&1/2\\-2&2&3&-1\\-4&1&3&4\end{array}\right][/tex]
- Row Operation 2: add 2 times the 1st row to the 2nd row
[tex]\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&3&1&0\\-4&1&3&4\end{array}\right][/tex]
- Row Operation 3: add 4 times the 1st row to the 3rd row
[tex]\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&3&1&0\\0&3&-1&6\end{array}\right][/tex]
- Row Operation 4: multiply the 2nd row by 1/3
[tex]\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&1&1/3&0\\0&3&-1&6\end{array}\right][/tex]
- Row Operation 5: add -3 times the 2nd row to the 3rd row
[tex]\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&1&1/3&0\\0&0&-2&6\end{array}\right][/tex]
- Row Operation 6: multiply the 3rd row by -1/2
[tex]\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&1&1/3&0\\0&0&1&-3\end{array}\right][/tex]
- Row Operation 7: add -1/3 times the 3rd row to the 2nd row
[tex]\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&1&0&1\\0&0&1&-3\end{array}\right][/tex]
- Row Operation 8: add 1 times the 3rd row to the 1st row
[tex]\left[\begin{array}{cccc}1&1/2&0&-5/2\\0&1&0&1\\0&0&1&-3\end{array}\right][/tex]
- Row Operation 9: add -1/2 times the 2nd row to the 1st row
[tex]\left[\begin{array}{cccc}1&0&0&-3\\0&1&0&1\\0&0&1&-3\end{array}\right][/tex]
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)
[tex]x_{1}= -3, x_{2} = 1, x_{3}= -3[/tex]
