Determinants
in Matlab
The determinant (in linear algebra)
is a value associated with a square matrix, that is a matrix with as
many rows as columns. It can be calculated from the elements of the
matrix by a specific arithmetic expression. The determinant provides
important information in many types of problems, for example, when the
matrix represent the coefficients of a system of linear
equations...
The
symbol
which consists
of the four numbers a_{1},
b_{1},
a_{2},
b_{2}
arranged in two rows and two columns is called a determinant of second order
or of order two.
The four numbers are called its elements. By definition,
Then,


Here, the elements 2 and 3 are in the first
row, and the
elements 4 and 1 are in the second row. Elements 2 and 4 are in column
one, and
elements 3 and 1 are column two.
The
method of solution of
linear equations by determinants is called the Cramer’s
Rule. A system of two linear equations in two unknowns may
be solved using a
second order det.
Given the system of equations
a_{1}x + b_{1}y = c_{1}
a_{2}x + b_{2}y = c_{2}
it is
possible to obtain
These
values for x
and y may
be written in terms of second order dets,
as
follows:
Example:
Solve
the system of equations:
2x
+ 3y
= 16
4x + y
= 3
The
denominator for both x and y is 

Then 

and 

In
Matlab, a determinant can be calculated with the builtin function
'det()'.
Using
the same numbers as in the example above,
if A =
[2 3; 4 1], then det(A) = 10;
if B =
[16 3; 3 1], then x =
det(A)/det(B)
= 2.5;
if C =
[2 16; 4 3], then y =
det(C)/det(A)
= 7
Naturally,
you can use the function det()
to find determinants of higher order.
Cramer's Rule Simultaneous
equations
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