Question

Solve the following system of equations
i. by using Cramer's rule and Matrix inversion method, when the coefficient matrix is nonsingular
ii. by using Gauss-Jordan method. Also determine whether the system has a unique solution or infinite number of solutions or no solution. Find the solution if exist.
x + y + z = 12x + 2y + 3z = 6x + 4y + 9z = 3

Answer 1

Answer:

 $x=\frac{-1}{8} ,y=\frac{39}{8},z=\frac{-7}{4}$  

Step-by-step explanation:

Analysis of equation: Since here  

$A=\left[\begin{array}{ccc}1&1&1\\12&2&3\\6&4&9\end{array}\right] \\\\\\X=\left[\begin{array}{ccc}x\\y\\z\end{array}\right] \\\\\\B=\left[\begin{array}{ccc}3\\3\\3\end{array}\right]$

Since rank of augmented matrix and coefficient are same,thus equations has consistent and hence has unique solution.

Cramer's Rule:

it says that

$x=\frac{\triangle}{\triangle_{1}} \\\\\\y=\frac{\triangle}{\triangle_{2}} \\\\\\x=\frac{\triangle}{\triangle_{3}}$

where Δ is determinant of matrix A,Δ1,Δ2,Δ3 are the determinant of A when column 1,2,3 are replaced by coefficient matrix respectively.$\triangle=\left|\begin{array}{ccc}1&1&1\\12&2&3\\6&4&9\end{array}\right|=-48\\\\\\\triangle_{1}=\left|\begin{array}{ccc}3&1&1\\3&2&3\\3&4&9\end{array}\right|=6\\\\\triangle_{2}=\left|\begin{array}{ccc}1&3&1\\12&3&3\\6&3&9\end{array}\right|=-234\\\\\\\triangle_{3}=\left|\begin{array}{ccc}1&1&3\\12&2&3\\6&4&3\end{array}\right|=84$

$x=\frac{-1}{8} ,y=\frac{39}{8},z=\frac{-7}{4}$  

2) Matrix inversion Method: As given matrix equations can be written as

AX=B

$X =A^{-1}.B$

$A^{-1}=  \left[\begin{array}{ccc}\frac{-1}{8}&\frac{-1}{48}&\frac{5}{48}\\\\\frac{-15}{8}&\frac{-1}{16}&\frac{-3}{16}\\\\\frac{-3}{4}&\frac{-1}{24}&\frac{5}{24}\end{array}\right] \\\\\\\\X=A^{-1}B=\left[\begin{array}{ccc}\frac{-1}{8}&\frac{-1}{48}&\frac{5}{48}\\\\\frac{-15}{8}&\frac{-1}{16}&\frac{-3}{16}\\\\\frac{-3}{4}&\frac{-1}{24}&\frac{5}{24}\end{array}\right] \times \left[\begin{array}{ccc}3\\3\\3\end{array}\right]$

$X=\left[\begin{array}{ccc}\frac{-1}{8}\\\\\frac{39}{8}\\\\\frac{-7}{4}\end{array}\right]$

 $x=\frac{-1}{8} ,y=\frac{39}{8},z=\frac{-7}{4}$  

3) Gauss -Jordan elimination method:In this method we write both augmented and coefficient matrix in metrix and convert the augmented matrix into row Echelon form,thus the coefficient matrix converted into the solution of the x,y,z variables

Reduce augmented matrix into row Echelon form by elemantary row operations

$\left[\begin{array}{ccc}1&1&1\\12&2&3\\6&4&9\end{array}\right] =\left[\begin{array}{ccc}3\\3\\3\end{array}\right]\\\\\\\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right] =\left[\begin{array}{ccc}\frac{-1}{8}\\\\\frac{39}{8}\\\\\frac{-7}{4}\end{array}\right]$