Given the system of equations 2x1 x2 x3 4 4x1 3x2 x3 6

Given the system of equations 2x_1 - x_2 + x_3 =4 4x_1 + 3x_2 - x_3 = 6 3x_1 + 2x_2 + 2x_3 = 15 Use Gauss-Jordan elimination to solve for the x\'_s. Compute the determinant. Substitute your results back into the original equations to check your solution.

Solution

In MATLAB, to solve Ax=b we begin by augmenting [A b]:

>> A=[2 -1 3; 4 3 -1;3 2 2]

A =

2 -1 3
4 3 -1
3 2 2

In MATLAB, we can find the inverse of a matrix using rref and then check that using the inv function: >> rref([A eye(size(A))])

rref([A eye(size(A))])

ans =

1.0000e+000 0 0 3.3333e-001 3.3333e-001 -3.3333e-001
0 1.0000e+000 0 -4.5833e-001 -2.0833e-001 5.8333e-001
0 0 1.0000e+000 -4.1667e-002 -2.9167e-001 4.1667e-001

>> inv(A)

ans =

3.3333e-001 3.3333e-001 -3.3333e-001
-4.5833e-001 -2.0833e-001 5.8333e-001
-4.1667e-002 -2.9167e-001 4.1667e-001

>> inv(A)*b

ans =

-1.6667e+000
5.6667e+000
4.3333e+000

these valuse substute in equations

>> 2*(-1.6667e+000)-1*5.6667e+000+3*4.3333e+000

ans =

3.9998e+000

>> 4*(-1.6667e+000)+3*5.6667e+000-1*4.3333e+000

ans =

6.0000e+000

3*(-1.6667e+000)+2*5.6667e+000+2*4.3333e+000

ans =

15.000e+000

We get exact values