Introduction to linear equations
- When we try to solve a set of linear equations we tend to take each equation separately and multiple or divide it so that we isolate and solve each variable.
- This method can be used to solve equations, but it can get too complicated when there are many equations with many variables.
- In such cases matrices can also be used to solve such linear equations.
It is required to have understood the following topics before proceeding with the lesson
Consider the set of linear equations:
\[\Large
a_{11}x_{1} + a_{12}x_{2} + a_{13}x_{3} + …….. + a_{1n}x_{n} = b_{1}\]
\[\Large a_{21}x_{1} + a_{22}x_{2} + a_{23}x_{3} + …….. + a_{2n}x_{n} = b_{2}\]
\[\Large ⋮ ~~~~~~~~~~~~~~~⋮~~~~~~~~~~~~~⋮~~~~~~~~~~~~~~~~~⋮~~~~~~~~~⋮ \]
\[\Large a_{n1}x_{1} + a_{n2}x_{2} + a_{n3}x_{3} + …….. + a_{nn}x_{n} = b_{n}
\]
From our knowledge of matrix multiplication, the above set of equations can be written in matrix form :
\[\Large
\begin{bmatrix}
a_{11} + a_{12}+ a_{13}…….. + a_{1n} \\
a_{21} + a_{22}+ a_{23}…….. + a_{2n} \\
⋮~~~~~~~⋮~~~~~~⋮~~~~~~~~~~~~~~⋮ \\
a_{n1} + a_{n2}+ a_{n3}…….. + a_{nn}
\end{bmatrix} ·
\begin{bmatrix}
x_{1} \\
x_{2} \\
⋮ \\
x_{n}
\end{bmatrix} =
\begin{bmatrix}
b_{1} \\
b_{2} \\
⋮ \\
b_{n}
\end{bmatrix}
~~~~i.e~ A·x = b
\]
If we multiply both sides of the matrix equation by the inverse of A, we have:
\[\Large
A^{-1}· Ax = A^{-1} · b
\]
\[\Large
But~ A^{-1}· A = I~~~~\therefore \,~I·x=A^{-1}·b~~~~~~i.e.~~x= A^{-1}·b
\]
Therefore, if we form the inverse of the matrix of coefficients and multiply matrix b by it, we shall determine the matrix of the solutions of x.
Only the inverse of matrix A must be found seperately to solve a set of linear equations?