Solving System of Linear Equations Using Cramer's Rule

1. Cramer's Rule can be used for
   1. Finding Solutions for Non-Homogeneous System of Linear Equations when Number of Equations is Same as Number of Variables.
   2. Finding whether the Homogeneous System of Linear Equations having Number of Equations Same as Number of Variables has only the Trivial Solution or has other Non-Trivial Solutions.
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
   
   \(A=\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{n1} & a_{n2} & ... & a_{nn}\end{bmatrix} \hspace{.5cm} X=\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} \hspace{.5cm} K=\begin{bmatrix} k_1 \\ k_2 \\ \vdots \\ k_n \end{bmatrix}\)
3. Cramer's Rule can be used for solving System of Linear Equations as given above using following steps
   1. Calculate Determinant \(D_0\) for Matrix \(A\) as the following
      
      \(D_0=\begin{vmatrix}a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{n1} & a_{n2} & ... & a_{nn}\end{vmatrix}\)
   2. Calculate Determinants \(D_1\), \(D_2\), ..., \(D_n\) by replacing the elements of \(n^{th}\) column of Matrix \(A\) with elements of Column Matrix \(K\) as the following
      
      \(D_1=\begin{vmatrix} k_1 & a_{12} & ... & a_{1n}\\ k_2 & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\k_n & a_{n2} & ... & a_{nn}\end{vmatrix}, \hspace{.5cm} D_2=\begin{vmatrix} a_{11} & k_1 & ... & a_{1n}\\ a_{21} & k_2 & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{n1} & k_n & ... & a_{nn}\end{vmatrix}, \hspace{.5cm}..., \hspace{.5cm} D_n= \begin{vmatrix} a_{11} & a_{12} & ... & k_1\\ a_{21} & a_{22} & ... & k_2 \\ \vdots & \vdots & \vdots & \vdots \\a_{n1} & a_{n2} & ... & k_n\end{vmatrix}\)
   3. For Homogeneous System of Linear Equations following can be the outcome
      1. If \(D_0\neq0\), the System of Linear Equations has only the Trivial Solution, i.e. values of all Variables \(x_1=x_2= ...= x_n=0\).
      2. If \(D_0=0\), then Infinite Number of Non-Trivial Solutions exist for the System of Linear Equations which can be found out by doing Elementary Row Operations on Matrix \(A\).
   4. For Non-Homogeneous System of Linear Equations following can be the outcome
      1. If all \(D_0=D_1=D_2=...= D_n=0\), then the System of Linear Equations has Infinitely Many Solutions.
      2. If \(D_0=0\) and atleast one of \(D_1\), \(D_2\), ..., \(D_n\) is Not Equal to 0, then the System of Linear Equations has No Solutions. Such a system is also called Inconsistent.
      3. If \(D_0\neq0\) then the System of Linear Equations has One Unique Solution. The values of the Variables \(x_1\), \(x_2\), ..., \(x_n\) can be found as follows
         
         \(x_1=\frac{D_1}{D_0},\hspace{.5cm}x_2=\frac{D_2}{D_0},\hspace{.5cm}...,\hspace{.5cm}x_n=\frac{D_n}{D_0}\)