Homogeneous System of Linear Equations

System of Linear Equation

A system of linear equations are group of $n$ linear equations containing $n$ number of variables.

1. System of 2 Linear Equations:

It is a pair of linear equations in two variables. It is usually of the form

$a_1x +b_1y + c_1 = 0$

$a_2x +b_2y + c_2 = 0$

Finding a solution for this system means finding the values of $x$ and $y$ that satisfy both equations.

2. System of 3 Linear Equations:

It is a group of 3 linear equations in three variables. It is usually of the form

$a_1x +b_1y + +c_1z + d_1 = 0$

$a_2x +b_2y + +c_2z + d_2 = 0$

$a_3x +b_3y + +c_3z + d_3 = 0$

Finding a solution for this system means finding the values of $x, y$, and $z$ that satisfy all three equations.

Homogenous System of Linear Equation

A linear equation with a constant value of zero is called a homogeneous equation. $\\\mathrm{Let,} \\\mathrm{a_1x+b_1y +c_1z=0\;\;\; ...(i)} \\\mathrm{a_2x+b_2y +c_2z=0\;\;\; ...(ii)} \\\mathrm{a_3x+b_3y +c_3z=0\;\;\; ...(iii)} \\\mathrm{be \; three\; homogeneous\; equations} \\\\\mathrm{and \; let\; \Delta = \begin{vmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{vmatrix}}$

Note that $x=y=z=0$ will always satisfy this system of equations. So system of homogeneous equations will always have at least one solution.

Also, the solution $x=0, y=0$, and $z=0$ is called a trivial solution, and other solutions are called non-trivial solutions.

• If $\Delta \neq 0$, then $x=0, y=0, z=0$ is the only solution of the above system. This solution is also known as a trivial solution.
• If $\Delta=0$, at least one of $x, y$, and $z$ are non-zero. In this case, we will have non-trivial solutions as well. Also, there would be infinite solutions of such a system of equations.

$\\\mathrm{Let,} \\\mathrm{a_1x+b_1y +c_1z=0\;\;\; ...(i)} \\\mathrm{a_2x+b_2y +c_2z=0\;\;\; ...(ii)} \\\mathrm{a_3x+b_3y +c_3z=0\;\;\; ...(iii)} \\\mathrm{be \; three\; homogeneous\; equation} \\\mathrm{and \; let\; \Delta = \begin{vmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{vmatrix}}$

If $\Delta\neq 0$, then $x=0, y=0, z=0$ is the only solution of the above system. This solution is also known as a trivial solution.

If $\Delta=0$, at least one of $x, y$ and $z$ are non-zero. This solution is called a non-trivial solution.

Explanation: using equation (ii) and (iii), we have

$
\begin{aligned}
&\begin{gathered}
\frac{x}{b_2 c_3-b_3 c_2}=\frac{y}{c_2 a_3-c_3 a_2}=\frac{z}{a_2 b_3-a_3 b_2} \\
\text { or } \frac{x}{\left|\begin{array}{ll}
b_2 & c_2 \\
b_3 & c_3
\end{array}\right|}=\frac{y}{\left|\begin{array}{ll}
c_2 & a_2 \\
c_3 & a_3
\end{array}\right|}=\frac{z}{\left|\begin{array}{ll}
a_2 & b_2 \\
a_3 & b_3
\end{array}\right|}=k \quad(\text { say } k \neq 0) \\
\therefore x=k\left|\begin{array}{ll}
b_2 & c_2 \\
b_3 & c_3
\end{array}\right|, \quad y=k\left|\begin{array}{ll}
c_2 & a_2 \\
c_3 & a_3
\end{array}\right|, \quad \text { and } \quad z=k\left|\begin{array}{ll}
a_2 & b_2 \\
a_3 & b_3
\end{array}\right|
\end{gathered}\\
&\text { Putting these values in equation (i), we have }\\
&a_1\left\{k\left|\begin{array}{ll}
b_2 & c_2 \\
b_3 & c_3
\end{array}\right|\right\}+b_1\left\{k\left|\begin{array}{ll}
c_2 & a_2 \\
c_3 & a_3
\end{array}\right|\right\}+c_1\left\{k\left|\begin{array}{ll}
a_2 & b_2 \\
a_3 & b_3
\end{array}\right|\right\}=0
\end{aligned}$

$\\\mathrm{\Rightarrow a_1\begin{vmatrix} b_2 &c_2 \\ b_3 & c_2 \end{vmatrix}-b_1\begin{vmatrix} a_2 & c_2\\ a_3 &c_3 \end{vmatrix}+c_1\begin{vmatrix} a_2 &b_2 \\ a_3 &b_3 \end{vmatrix} = 0 \;\;\;[\because \; k \neq 0]} \\\mathrm{or \;\; \begin{vmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{vmatrix} = 0 \; or \; \Delta = 0}$

This is the condition for a system to have a Non-trivial solution.

Non-Homogenous System of Linear Equations

A linear equation with a constant value not equal to zero is called a homogeneous equation.

Characteristics of non Homogenous Linear Equations

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

Recommended Video Based on Homogeneous System of Linear Equations:

Solved Examples Based on Homogenous System of Linear Equations

Example 1:

Solution

For non-trivial solution $\Delta = 0$

$\Rightarrow \begin{vmatrix} 1+\cos ^{2}\theta& \sin ^{2}\theta &4\sin 3\theta \\ \cos ^{2}\theta&1+\sin ^{2}\theta &4\sin 3\theta \\ \cos ^{2}\theta&\sin ^{2}\theta & 1+4\sin 3\theta \end{vmatrix}= 0$

$R_{3}\rightarrow R_{3}-R_{2},R_{2}\rightarrow R_{2}-R_{1}$

$\Rightarrow \begin{vmatrix} 1+\cos ^{2}\theta& \sin ^{2} \theta&4\sin 3\theta \\ -1& 1&0 \\ 0 & -1 & 1 \end{vmatrix}= 0$
$\Rightarrow \left ( 1+\cos ^{2}\theta \right )-\sin ^{2} \theta\left ( -1 \right )+4\sin 3\theta= 0$

$\Rightarrow 2+4 \sin 3\theta= 0$
$\Rightarrow \sin 3\theta= \frac{-1}{2}$
$\Rightarrow 3\theta= \frac{7\pi}{6},\frac{11\pi}{6}$
$\Rightarrow \theta= \frac{7\pi}{18},\frac{11\pi}{18}$
$But\: \frac{11\pi}{18}\not\in\left ( 0,\frac{\pi}{2} \right )$

$\Rightarrow \theta= \frac{7\pi}{18}$

Example 2 :

Solution

$\alpha +\beta +\gamma =2\pi$

$\\ \Delta =\begin{vmatrix} 1 & \cos\gamma &\cos \beta \\ \cos\gamma & 1 & \cos \alpha \\ \cos\beta & \cos \alpha & 1 \end{vmatrix}\\$

$=1+2\cos\alpha \cos\beta \cos\gamma -\cos^{2}\beta -\cos^{2}\alpha -\cos^{2}\gamma \\$

$Let \; \gamma =2\pi-\alpha -\beta \\$

$\Rightarrow \Delta =1+2\cos\alpha \cos\beta \cos\left ( \alpha +\beta \right )-\cos^{2}\left ( \alpha +\beta \right )\\$

$=1+2\cos\alpha \cos\beta \left [ \cos\alpha \cos\beta -\sin\alpha \sin\beta \right ]-\cos^{2}\alpha -\cos^{2}\beta\\$

$-\left [ \cos\alpha \cos\beta -\sin\alpha \sin\alpha \right ]^{2}\\$

$=1+2\cos^{2}\alpha \cos^{2}\beta -2\sin\alpha \sin\beta \cos\alpha \cos\beta -\cos^{2}\alpha -\cos^{2}\beta \\$

$-\cos^{2}\alpha \cos^{2}\beta -\sin^{2}\alpha \sin^{\beta }+2\sin\alpha \sin\beta \cos\alpha \cos\beta \\$

$= 1-\cos^{}\alpha -\cos^{\beta }+\cos^{2}\alpha \cos^{2}\beta -\sin^{2}\alpha \sin^{2}\beta \\$

$=\left ( 1-\cos^{2}\alpha \right )\left ( 1-\cos^{2}\beta \right )-\sin^{2}\alpha \sin^{}\beta \\$

$=\sin^{2}\alpha \sin^{2}\beta -\sin^{2}\alpha \sin^{2}\beta=0$

Example 3: