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Mathematical Expression Editor
Octave for Chapter 7
Examples in this section provide sample Octave code for finding determinants and
illustrating their uses and properties. You can access our code through the link at the
bottom of each template. Feel free to modify the code and experiment to learn
more!
You can write your own code using Octave software or online Octave cells. To access
Octave cells online, go to the Sage Math Cell Webpage, select OCTAVE as the
language, enter your code, and press EVALUATE.
To ”save" or share your online code, click on the Share button, select Permalink, then
copy the address directly from the browser window. You can store this link to access
your work later or share this link with others. You will need to get a new Permalink
every time you modify the code.
Octave Tutorial
In practice, computing determinants using Laplace Expansion is computationally
expensive. \(LU\)-factorization can be used to improve efficiency, but may be numerically
unstable. As a result, determinants are rarely used in computational practice.
Examples and exercises in this section are designed to illustrate theoretical properties
and computational pitfalls associated with determinants and should not be
interpreted as templates for large-scale computational problems.
We will use the det function to find determinants in Octave.
Compute the determinant of each of the following matrices. If a matrix is singular,
say so.
Illustrate part (b) of the above theorem using the second row of matrix \(A\), and
constant \(k=4\).
We will use the following code to illustrate the property.
% Define rows of matrix A
r_1=[2 -1 5];
r_2=[0 3 -4];
r_3=[-2 1 -1];
% Combine the rows to form matrix A
A=[r_1; r_2; r_3];
% Compute the determinant of A
det_A=det(A)
% Form matrix B by multiplying the second row of A by k=4
B=[r_1; 4*r_2; r_3];
% Compute the determinant of B
det_B=det(B)
% Compute 4*det(A)
det_A_times4=4*det(A)
For our next example, recall the following theorem, known as Cramer’s Rule.
?? Let \(A\) be a nonsingular \(n\times n\) matrix, and let \(\vec {b}\) be an \(n\times 1\) vector. Then the components of the
solution vector \(\vec {x}\) of \(A\vec {x}=\vec {b}\) are given by
\[x_i=\frac {\det {A_i(\vec {b})}}{\det {A}}\]
where \(A_i(\vec {b})\) is matrix \(A\) with its \(i^{th}\) column replaced by \(\vec {b}\).
Use Octave to solve the following system for \(x_1\) using Cramer’s rule.
% Define columns of matrix A
c_1=[1; -1; 0];
c_2=[2; 1; 3];
c_3=[-1; 1; 1];
% Combine the columns to form matrix A
A=[c_1 c_2 c_3];
% Define vector b
b=[-2; 3; 1];
% Form A_1 by replacing the first column of A with b
A_1=[b c_2 c_3];
% Find x_1
x_1=det(A_1)/det(A)
Determine whether each of the above matrices is singular in two different ways.
Use the det function
Use an alternative approach
Are the two methods used above in agreement? Click the arrow on the right to learn
more.
Octave uses \(LU\)-decomposition to compute the determinant. This method is susceptible
to rounding errors ( Reference). Because of this, using the det function to determine
singularity is generally not a good idea.
If \(A\) is non-singular, find \(A^{-1}\), and \(\det (A^{-1})\). Illustrate the relationship between \(\det (A)\) and \(\det (A^{-1})\)
computationally.
Modify Example 7 to illustrate parts (a) and (c) of Theorem ??. (Use rows and
coefficients of your choice.)
Modify the code of Example 10 to find \(x_2\) and \(x_3\).