Solving Linear Equations Using Matrix - Practice Questions & MCQ

Let us consider n linear equations in n unknowns, given as below

$\begin{array}{rl}& {\mathrm{a}}_{11}{\mathrm{x}}_1+{\mathrm{a}}_{12}{\mathrm{x}}_2+\dots +{\mathrm{a}}_{1\mathrm{n}}{\mathrm{x}}_{\mathrm{n}}={\mathrm{b}}_1\\ & {\mathrm{a}}_{21}{\mathrm{x}}_1+{\mathrm{a}}_{22}{\mathrm{x}}_2+\dots +{\mathrm{a}}_{2\mathrm{n}}{\mathrm{x}}_{\mathrm{n}}={\mathrm{b}}_2\\ & \text{... ... ... ... ... ... }\\ & \text{... ... ... ... ... ... }\\ & {\mathrm{a}}_{\mathrm{n}1}{\mathrm{x}}_1+{\mathrm{a}}_{\mathrm{n}2}{\mathrm{x}}_2+\dots +{\mathrm{a}}_{\mathrm{nn}}{\mathrm{x}}_{\mathrm{n}}={\mathrm{b}}_{\mathrm{n}}\end{array}$
Here ${\mathrm{x}}_1,{\mathrm{x}}_2,\dots {\mathrm{x}}_{\mathrm{n}}$ are n unknown variables
if $b_1=b_2=\dots =b_n=0$ then the system of equation is known as homogenous system of equation and if any of $b_1,b_2,\dots b_n$ is non - zero then it is called non homogenous system of equation

The above system of equations can be written in matrix form as

$\left[\begin{array}{ccccc}{a}_{11}& a_{12}& \dots & \dots & a_{1n}\\ a_{21}& a_{22}& \dots & \dots & a_{2n}\\ \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots \\ a_{n1}& a_{n2}& \dots & \dots & a_{nn}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ \dots \\ \dots \\ x_n\end{array}\right]=\left[\begin{array}{c}{b}_1\\ b_2\\ \dots \\ \dots \\ b_n\end{array}\right]$

$\Rightarrow \mathrm{AX}=\mathrm{B}$, where

$\mathrm{A}=\left[\begin{array}{ccccc}{a}_{11}& a_{12}& \dots & \dots & a_{1n}\\ a_{21}& a_{22}& \dots & \dots & a_{2n}\\ \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots \\ a_{n1}& a_{n2}& \dots & \dots & a_{nn}\end{array}\right],\mathrm{X}=\left[\begin{array}{c}{x}_1\\ x_2\\ \dots \\ \dots \\ x_n\end{array}\right],\mathrm{B}=\left[\begin{array}{c}{b}_1\\ b_2\\ \dots \\ \dots \\ b_n\end{array}\right]$

Premultiplying equation $AX=B$ by $A^{-1}$, we get

$\begin{array}{rl}& A^{-1}(AX)=A^{-1}B\Rightarrow \left(A^{-1}A\right)X=A^{-1}B\\ & \Rightarrow IX=A^{-1}B\\ & \Rightarrow X=A^{-1}B\end{array}$
$X=\frac{\mathrm{adj}A}{|A|}B$
Types of equation:
1. System of equations is non-homogenous:

If $|A|\ne 0$, then the system of equations is consistent and has a unique solution $X=A^{-1}B$
If $|A|=0$ and $(\mathrm{adj}A)\cdot B\ne 0$, then the system of equations is inconsistent and has no solution.
If $|A|=0$ and $(\mathrm{adj}A)\cdot B=0$, then the system of equations is consistent and has infinite number of solutions.
1. System of equations is homogenous:

If $|A|\ne 0$, then the system of equations has only one solution which is the trivial solution.
If $|\mathrm{A}|=0$, then the system of equations has non-trivial solution and it has an infinite number of solutions.