curves he considers, Cramer presents effective techniques for drawing their graphs. In the second chapter
he discusses the role of geometric transformations as a
means of simplifying the equations to curves (akin to
today’s approach using the notion of PRINCIPAL AXES).
This leads to his famous classification of curves in the
third chapter. Here Cramer also discusses the problem
of finding the equation of a degree-two curve ax
2 + bxy
+ cy
2 + dx + ey = 0 that passes through five previously
specified points in the plane. Substituting in the values
of those points leads to five linear equations in the five
unknowns a, b, c, d, and e. To solve the problem
Cramer then refers the reader to an appendix of the
text, and it is here that his famous rule for solving systems of equations appears. Cramer made no claim to
the originality of the result and may have been well
aware that Colin Maclaurin had first established the
famous theorem. (Cramer cited the work of Colin
Maclaurin in many footnotes throughout his text, suggesting that he was working closely with the writings
of Maclaurin.)
Cramer also served in local government for many
years, offering expert opinion on matters of artillery
and defense, excavations, and on the reconstruction
and preservation of buildings. He died on January 4,
1752, in Bagnols-sur-Cèze, France. Although Cramer
did not invent the rule that bears his name, he deserves
recognition for developing superior notation for the
rule that clarified its use.
Cramer’s rule Discovered by Scottish mathematician
COLIN MACLAURIN (1698–1746), but first published
by Swiss mathematician GABRIEL CRAMER (1704–52),
Cramer’s rule uses the DETERMINANT function to find a
solution to a set of SIMULTANEOUS LINEAR EQUATIONS.
An example best illustrates the process.
Consider the set of equations:
2x + 3y + z = 3
x – 2y + 2z = 11
3x + y – 2z = –6
Set A to be the matrix of coefficients:
The determinant of this matrix is not zero: det(A) = 35.
A standard property of determinants asserts that
if the elements of the first column are multiplied by
the value x, then the determinant changes by the factor x:
Adding a multiple of another column to the first does
not change the value of the determinant. We shall add y
times the second column, and z times to the third to
this first column:
But of course this first column equals the column of
values of the simultaneous equations:
This tells us that the value of x we seek is:
In the same way, the value y is found as the ratio of the
determinant of the matrix A with the second column
replaced by the column of values of the simultaneous
equations and the determinant of A:
y
A
=
− −
=
− = −
det
det( )
2 3 1
1 11 2
3 6 2
35
35
1
x
A
=
=
=
35
35
35
1
det( )
x
A
x y z
x
y
z
x y
z
det( ) det
det
=
+ +
− +
−
+ −
−
=
−
−
−
=
2
3
3 1
2 2
2 2
3
2
1
2
3 3 1
11 2 2
6 1
2
35
det
det( )
2
3
3 1
2 2
2 2
3
2
1
2
x y z
x
y
z
x y
z
x
A
+ +
− +
−
+ −
−
=
det
det( )
2
3 1
2 2
3
1
2
x
x
x
x
A
−
−
=
A =
−
−
2 3 1
1 2 2
3 1
2
108 Cramer’s rule
he discusses the role of geometric transformations as a
means of simplifying the equations to curves (akin to
today’s approach using the notion of PRINCIPAL AXES).
This leads to his famous classification of curves in the
third chapter. Here Cramer also discusses the problem
of finding the equation of a degree-two curve ax
2 + bxy
+ cy
2 + dx + ey = 0 that passes through five previously
specified points in the plane. Substituting in the values
of those points leads to five linear equations in the five
unknowns a, b, c, d, and e. To solve the problem
Cramer then refers the reader to an appendix of the
text, and it is here that his famous rule for solving systems of equations appears. Cramer made no claim to
the originality of the result and may have been well
aware that Colin Maclaurin had first established the
famous theorem. (Cramer cited the work of Colin
Maclaurin in many footnotes throughout his text, suggesting that he was working closely with the writings
of Maclaurin.)
Cramer also served in local government for many
years, offering expert opinion on matters of artillery
and defense, excavations, and on the reconstruction
and preservation of buildings. He died on January 4,
1752, in Bagnols-sur-Cèze, France. Although Cramer
did not invent the rule that bears his name, he deserves
recognition for developing superior notation for the
rule that clarified its use.
Cramer’s rule Discovered by Scottish mathematician
COLIN MACLAURIN (1698–1746), but first published
by Swiss mathematician GABRIEL CRAMER (1704–52),
Cramer’s rule uses the DETERMINANT function to find a
solution to a set of SIMULTANEOUS LINEAR EQUATIONS.
An example best illustrates the process.
Consider the set of equations:
2x + 3y + z = 3
x – 2y + 2z = 11
3x + y – 2z = –6
Set A to be the matrix of coefficients:
The determinant of this matrix is not zero: det(A) = 35.
A standard property of determinants asserts that
if the elements of the first column are multiplied by
the value x, then the determinant changes by the factor x:
Adding a multiple of another column to the first does
not change the value of the determinant. We shall add y
times the second column, and z times to the third to
this first column:
But of course this first column equals the column of
values of the simultaneous equations:
This tells us that the value of x we seek is:
In the same way, the value y is found as the ratio of the
determinant of the matrix A with the second column
replaced by the column of values of the simultaneous
equations and the determinant of A:
y
A
=
− −
=
− = −
det
det( )
2 3 1
1 11 2
3 6 2
35
35
1
x
A
=
=
=
35
35
35
1
det( )
x
A
x y z
x
y
z
x y
z
det( ) det
det
=
+ +
− +
−
+ −
−
=
−
−
−
=
2
3
3 1
2 2
2 2
3
2
1
2
3 3 1
11 2 2
6 1
2
35
det
det( )
2
3
3 1
2 2
2 2
3
2
1
2
x y z
x
y
z
x y
z
x
A
+ +
− +
−
+ −
−
=
det
det( )
2
3 1
2 2
3
1
2
x
x
x
x
A
−
−
=
A =
−
−
2 3 1
1 2 2
3 1
2
108 Cramer’s rule