Respuesta :
Answer:
Step-by-step explanation:
There are different ways to solve linear systems, it mean, to find if there exist a unique solution, multiple solutions or no solution. For simple systems we can use algebraic methods like equality, substitution or reduction methods.
Another way to solve linear equation systems is using matrix. To do this we have to express the equation systems in terms of AX = B, where A is the coefficients matrix, X is the unknowns matrix and B is the independent coefficients matrix.
For example, the following system
5x - 3y + z = 10
x + y -z = 1
x + z =6
Can be expressed as:
[tex]\left[\begin{array}{ccc}5&-3&1\\1&1&-1\\1&0&1\end{array}\right][/tex]·[tex]\left[\begin{array}{ccc}x\\y\\z\end{array}\right][/tex] =[tex]\left[\begin{array}{ccc}10\\1\\6\end{array}\right][/tex]
To find if there is solution or not, we can apply the inverse matrix method or the Gauss elimination method. The first one consist in multiplying both parts of the equality by the inverse coeficient´s matrix. So, X = (A^-1)·B.
In the Gauss elimination method, we form the augmented matrix [A|B]. In terms of the previous example:
[tex]\left[\begin{array}{ccc}5&-3&1|10\\1&1&-1|1\\1&0&1|6\end{array}\right][/tex]
We must apply elemental operations on rows until get a diagonal matrix. Depending on the solution obtained for the last row, the system will be compatible or not:
- If we found that the last row has the form 0x + 0y + cz = k, with k≠0, the system has only one solution.
- If we found that the last row has the form 0x + 0y + 0z = k, the system has no solution.
- If we found that the last row has the form 0x + 0y + 0z = k, the system has multiple solutions.
The rank of a matrix is defined as the number of rows (or columns) that are linearly independent. A linear system has one or multiple solutions if the rank of the augmented matruix is the same of matrix A. If it is not the case, the system has no solution.
