Section 4.3 Matrix Equations
We can represent a system of equations as a matrix equation by writing \(Ax = b\) where \(A\) is the coefficient matrix, \(x\) is the vector of variables, and \(b\) is the vector of constants.
\begin{align*}
\begin{bmatrix}
a_{11} & a_{12} & \dots & a_{1n} \\
a_{21} & a_{22} & \dots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \dots & a_{mn}
\end{bmatrix}
\begin{bmatrix}
x_{1} \\
x_{2} \\
\vdots \\
x_{n}
\end{bmatrix}
\hspace{2pt}
=
\hspace{2pt}
\begin{bmatrix}
b_{1} \\
b_{2} \\
\vdots \\
b_{m}
\end{bmatrix}
\end{align*}
We can solve a matrix equation in Sage using the method
solve_right(). Evaluating A.solve_right(b) returns a vector \(x\) such that \(Ax = b\text{.}\) The components of this vector are the values of a solution of the system of equations.
Subsection 4.3.1 Case I
Now, let’s solve the \(\textbf{Case I}\) system applying the matrix equation method.
Subsection 4.3.2 Case II
Now, let’s solve the \(\textbf{Case II}\) system applying the matrix equation method.
We found that one solution to the system is \(x = 4,\ y = 0\text{,}\) and \(z = 0\text{.}\) This is just one particular solution when the parameter is zero. Observe that we do not obtain all the infinitely many solutions of the system using this method.
Subsection 4.3.3 Case III
Now, let’s solve the \(\textbf{Case III}\) system applying the matrix equation method.
Observe that in this case, Sage produces an error message. Nevertheless, the error message states that the system has no solutions.
To avoid the error message and continue the execution of the program, we can write:
