
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 x_{1},
x_{2}, . . . x_{n},
the coefficients (a_{ij})
and constants (b_{i})
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 (a_{ij}), 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^{1}Ax =
A^{1}b 
since,
A^{1}A =
I by definition and Ix =
x, where I
is the identity matrix, as 

then 
x =
A^{1}b. 
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 equations
using the inverse matrix method.


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^{1}A =
I and Ix =
x, then 

Therefore,
the solution is, x_{1}
= 1, x_{2} = 2
and x_{3}
= 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 = a_{ik} is
nonzero, and in this
case, the value of the unknown x_{k}
is given by the
expression 

where
the numerator A_{k}
is equal to the determinant A
of the matrix A
that results when the k^{th}
column (the coefficients
of the unknown to be found) of the coefficient matrix is
replaced by the column of constants, b_{1},
b_{2},
. . . , b_{n}. 
Let
show that Cramer's Rule, to find the value of the unknown x_{k},
coincide with the solution given by the matrix
equation x =
A^{1}b. 
That
is, by expanding the determinant A_{k}
by the k^{th}
column we get 
A_{k} = b_{1}A_{1}_{k}
+ b_{2}A_{2}_{k}
+ . . . + b_{n}A_{n}_{k} 
where, A_{1}_{k}, A_{2}_{k},
. . . , A_{n}_{k}
are the cofactors of the entries, b_{1},
b_{2},
. . . , b_{n},
that are the same as the 
cofactors
of the entries, a_{1}_{k},
a_{2}_{k},
. . . , a_{n}_{k},
of the determinant A =
det(A) or the matrix A. 
To
find the value of unknown x_{k}
from x =
A^{1}b
we should calculate the scalar product of the k^{th}
row vector of A^{1}
and the column vector b. 
Therefore,
if a_{i}_{k}
denotes the entries of A^{1}
then 
x_{k} =
a_{k}_{1}b_{1}
+ a_{k}_{2}b_{2}
+ . . . + a_{k}_{n}b_{n}. 
Recall
that 

where A_{ki}_{}
denotes the ki^{th}
entry of
the transpose of the cofactor matrix. 


As
the transposition interchange rows and columns, the above expression for the unknown x_{k}
can also be written as 

what
coincide with 

where A_{k} = b_{1}A_{1}_{k}
+ b_{2}A_{2}_{k}
+ . . . + b_{n}A_{n}_{k}. 


That
proves the correlation between solution given by Cramer's Rule
and the solution given by the matrix
equation x =
A^{1}b. 








Matrices, determinants and systems of linear equations
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