Application of Inverse Matrices in Linear Equation Systems

In this article we will produce more about linear equations and invertibility matrices. So we will find a new method for solving unknown unknowns.

Theorem 1.
If A is a matrix of n x n which can be reversed, then for each B matrix measuring n x 1, the system equation AX = B has exactly one solution, namely X = A-1B.

Proof:
Because A (A-1B) = B, then X = A-1B is a solution AX = B. To show that this is one solution, we will assume that X0 is any solution, and then show that X0 must be a solution of A-1B..
If X0 is a solution, then AX0 = B. By multiplying the two sections A-1, we get X0 = A-1B.

Example 1
Review the system of linear equations below,

  X1 + 2X2 + 3X3 = 3
2X1 + 5X2 + 3X3 = 3
  X1            + 8X3 = 17

In the form of a matrix, this system can be written as AX = B, where


Because the A-1 matrix is ​​known in Example 4 of the article "Elementary Line Operation Methods and Determinants for Finding the Inverse Matrix", in this article we only display the results of the A-1 matrix, namely,
According to the theorem 1 in this article, the completion of this system is
or x1 = 1, x2 = -1, x3 = 2.

Let A be a reversible matrix, then the theorem 1 solves this problem completely by specifying that for each B matrix measuring m x 1, the equation AX = B has a unique solution X = A-1B. If A is not a quadratic matrix, or if A is a quadratic matrix but cannot be reversed, then this 1 theorem cannot be applied. In these cases we want to determine what conditions, if there is something that must be met by the B Matrix so that AX = B is consistent. The following example illustrates how Gauss-Jordan elimination can be used to determine such conditions.

Example 2.
What conditions must be met by b1, b2, b3, so that the systems of the following equations are consistent?

  x1 + x2 + 2x3 = b1
  x1          + 2x3 = b2
2x2 + x2 + 3x3 = b3

Solution:
The enlarged matrix for the system is
Which can be reduced to the form of row echelon as follows,
It is clear now from the third row in the matrix that the system has a solution if and only if b1, b2, and b3 meet the conditions.

b3 - b2- b1 = 0 or  b3 = b1 + b2

to state this condition in another way, then AX = B is consistent if and only if B is a shaped matrix,
Where b1 and b2 are any.

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