Systems of Linear Equations
      Solving systems of equations using matrices
         Inverse matrix method
         Cramer’s rule
         Cramer’s Rule and inverse matrix method correlation
Systems of Linear Equations
Solving systems of equations using matrices
A system of linear equations is a set of n equations in n unknowns (variables) of the form
where, the unknowns are denoted by x1, x2, ... xn, the coefficients (aij) and constants (bi) are given values.
The system of equations above can be written in matrix form as
  or Ax = b  
where, A is matrix of the coefficients (aij), x is a vector of n unknowns and b is a vector of n constants. 
Inverse matrix method
The matrix equation  Ax = b,  where A is an n ´ n regular matrix (det(A) is not 0), multiplied by A-1 gives 
A-1Ax = A-1b
since,  A-1A = I by definition and  Ix = x,  where I is the identity matrix, as 
x = A-1b.
Thus, the inverse matrix method uses this matrix equation to find the solution to the system of equations directly.
Example:  Find the solution of the given three simultaneous equations using the inverse matrix method.
The given equations are written as the equivalent matrix equation. 
Then, both sides of the above matrix equation we multiply by the inverse of the coefficient matrix A (calculation of which is shown in the previous example),
Since. A-1A = I  and  Ix = x, then
Therefore, the solution is, x1 = 1, x2 = 2 and x3 = 3.
Cramer’s Rule and inverse matrix method correlation
Cramer's Rule says that a system of n linear equations in n unknowns, 
will have a unique solution if the determinant of the coefficient matrix det(A) = A = | aik | is nonzero, and in this case, the value of the unknown xk is given by the expression
where the numerator Ak is equal to the determinant A of the matrix A that results when the kth column (the coefficients of the unknown to be found) of the coefficient matrix is replaced by the column of constants, b1, b2, . . . , bn.
Let show that Cramer's Rule, to find the value of the unknown xk, coincide with the solution given by the matrix equation  x = A-1b.
That is, by expanding the determinant Ak by the kth column we get
Akb1A1k + b2A2k + . . . +  bnAnk
where, A1k, A2k, . . . , Ank are the cofactors of the entries, b1, b2, . . . , bn, that are the same as the cofactors of the entries, a1k, a2k, . . . , ank, of the determinant A = det(A) or the matrix A.
To find the value of unknown xk from x = A-1b we should calculate the scalar product of the kth row vector of A-1 and the column vector b.
Therefore, if aik denotes the entries of A-1 then
xk = ak1b1 + ak2b2 + . . . +  aknbn.
Recall that where Aki denotes the kith entry of the transpose of the cofactor matrix.  
As the transposition interchange rows and columns, the above expression for the unknown xk can also be written as
what coincide with   where Akb1A1k + b2A2k + . . . +  bnAnk.  
That proves the correlation between solution given by Cramer's Rule and the solution given by the matrix equation  x = A-1b.
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