1.4 KiB
#math/linear_algebra
matrix representation
#card a system can be represented by a matrix. each column represents the coefficients of each variable involved, the rows represent each equation in the system. Optionally, an augmented matrix includes a bar followed by the constant (but that's not there when the constant is 0) For a given system of equations
\begin{aligned}
x_{1}+3 x_{2}+2 x_{3}+3 x_{4} &=-4 \\
x_{2}-2 x_{3}-2 x_{4} &=3 \\
-x_{1}-3 x_{2}+2 x_{3}+x_{4} &=4
\end{aligned}
we can express it as a matrix of coefficients
\left[\begin{array}{rrrr}
1 & 3 & 2 & 3 \\
0 & 1 & -2 & -2 \\
-1 & -3 & 2 & 1
\end{array}\right]
or as an augmented matrix
\left[\begin{array}{rrrr|r}
1 & 3 & 2 & 3 & -4 \\
0 & 1 & -2 & -2 & 3 \\
-1 & -3 & 2 & 1 &
\end{array}\right]
!
^1651526044974
solve
You can do whatever you want with these elementary operations:
!
You wanna use those operations to reduce the matrix to row echelon form like so:
(reduced) row echelon form
!
these are often solutions to the system of equations. If a row and column intersection has just a one, then the little margin on the right signifies the solution for that column's corresponding variable.
the algorithm
the procedure to do so is as follows:
!