Systems of Linear Equations

Basically simultaneous equations. Can be written as $Ax=b$

The coefficient matrix $A$ is said incomplete matrix. The matrix $A^c\in \R_{m,n+1}$ whose first $n$ columns are those of the matrix $A$ and the $(n+1^{th})$ column is the vector $b$ is said complete matrix

$A^c=(A|b)=$

Cramer's Theorem

If $A$ is non-singular , the is only one solution simultaneously satisfying all the equations: if $detA\ne 0$ then $\exist!x$ such that $Ax=b$

Hybrid Matrix

Hybrid matrix with respect to the $i^{th}$ column is the matrix $A_i$ obtained from $A$ by substituting the $i^{th}$ column with $b$ (Switch $a^i$ with $b$)

Cramer's Method

For a given system of linear equations $Ax=b$ with $A$ non-singular, a generic solution $x_i$ element of $x$ can be computed as $x_i=\frac{detA_i}{detA}$

For a square matrices would repeat 3 times, change column 3 times with $b$