Solving System of Linear Equations Using Inverse of Matrix

1. Matrix Inversion can be used to Find Solutions for Non-Homogeneous System of Linear Equations when Number of Equations is Same as Number of Variables.
2. Given following System of Linear Equations
   
   \(a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n = k_1\)
   \(a_{21}x_1 + a_{22}x_2 + ... + a_{1n}x_n = k_2\)
   \(\vdots \)
   \(a_{n1}x_1 + a_{n2}x_2 + ... + a_{nn}x_n = k_{n}\)
   
   The Co-efficients, Variables, and the Constants of the above Equations can be represented by the following Matrices
   
   \(A=\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{n1} & a_{n2} & ... & a_{nn}\end{bmatrix} \hspace{.5cm} X=\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} \hspace{.5cm} K=\begin{bmatrix} k_1 \\ k_2 \\ \vdots \\ k_n \end{bmatrix}\)
   
   The solutions for the System of Linear Equations given above can be found using Matrix Inversion only if Determinant of Matrix \(A\) is Not Zero (i.e \(|A|\neq0\)).
3. Following are the steps for solving the System of Linear Equations using Matrix Inversion
   
   The System of Linear Equations can be represented using following Matrix Equation
   
   \(AX=K \Rightarrow \begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{n1} & a_{n2} & ... & a_{nn}\end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}=\begin{bmatrix} k_1 \\ k_2 \\ \vdots \\ k_n \end{bmatrix}\)   ...(1)
   
   Multiplying the Matrix Equation (1) with \(A^{-1}\) on both the sides we get
   
   \(A^{-1}AX=A^{-1}K\hspace{.6cm}\Rightarrow IX=A^{-1}K\hspace{.6cm}\Rightarrow X=A^{-1}K\)
   
   \(\Rightarrow \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}=\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{n1} & a_{n2} & ... & a_{nn}\end{bmatrix}^{-1}\begin{bmatrix} k_1 \\ k_2 \\ \vdots \\ k_n \end{bmatrix}\)   ...(2)
   
   Hence by Finding the Inverse of Matrix \(A\) and Multiplying it with the Vector \(K\), we can find the Vector \(X\), and hence the values of Variables \(x_1\), \(x_2\), ..., \(x_n\).