por Systems of equations are sets of equations inside which the same variables are housed.. The solutions to these systems of equations generally consist of assigning values to the variables so that they satisfy each equation. Its solution is given by finding all the solutions within the system.
What are systems of equations?
A system of m equations (as we see in the image, where we have a row, another row and others m rows) with n unknowns (which in the image are represented by the columns) can be written as follows:
As we have seen before, we can also write it in the following way, which is the matrix form:
We have, therefore, the first equation in the first row, the second equation in the second row, the third equation in the third row, and so on until the n rows.
This entire matrix is summarized as TO . X = B
We also have the column vector, which has the components of the vector x. Next, we have the resulting vector B.
The coefficient matrix is called:
And matrix expanded to:
Rewriting the system of equations
Here we are going to apply what we already know to rewrite it using a more compact form. To do this, we take into account the coefficient matrix:
Ax = B
Where A is a matrix of mxn, x and b are two column vectors of length n. In order to solve this equation Ax = B, we must solve for x:
Ax = B
A-1 Ax = A-1 B
Ix = A-1 B
x = A-1 B
Determine if systems of equations can be solved
Systems of linear equations can have:
A single solution → given compatible system
Infinite solutions → undetermined compatible system
No solution → incompatible system
Determined compatible system
The systems of linear equations will be determined compatible systems, that is, they will have a unique solution when the determinant of the matrix A is equal to 0.
det (A) ≠ 0 → Determined compatible system
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