Q4: For the linear system 

2x -  y + 3z =  5
3x +  y + 4z =  2
-x + 5y – 2z = -2 

(i)Write the system in matrix from AX = B 

    A      X =   B 

[ 2 -1  3][x]   [ 5]
[ 3  1  4][y] = [ 2]
[-1  5 –2][z]   [-2] 


(ii)Obtain the determinant and inverse of A. 
 
I will assume you know how to get the value 
of a 3×3 determinant, and how to find the
minors and the transpose. If you don't post 
again asking how.

         | 2 -1  3|           
det(A) = | 3  1  4| = 2           
         |-1  5 –2|           

First find the cofactor matrix.
Replace each element of A by the value of
its minor 2×2 determinant with the sign
left as it is or changed according to the
sign scheme:

       |+ - +|
       |- + -|
       |+ - +|

                  [-22   2   16]
cofactor matrix = [ 13  -1   -9]
                  [ -7   1    5]

Take the transpose of the cofactor matrix,
which is called the adjoint matrix:

                  [-22  13   -7]
adjoint matrix =  [  2  -1    1]
                  [ 16  -9    5]

Divide every member of the adjoint matrix
by the determinant of A.  This is the
inverse of A, written A-1:

                       [-11 13/2  -7/2]
Inverse matrix = A-1 = [  1 -1/2   1/2]
                       {  8 -9/2   5/2]   



(iii)Solve the system.

Go back to the system in matrix form AX = B

                      A     X  =   B

                 [ 2 -1  3][x]   [ 5]
                 [ 3  1  4][y] = [ 2]
                 [-1  5 –2][z]   [-2] 

Left-multiply both sides by the inverse matrix and
get the form A-1AX = A-1Bx

      A-1             A     X  =          A-1      X  
    
[-11 13/2  -7/2] [ 2 -1  3][x]   [-11 13/2  -7/2][ 5]
[  1 -1/2   1/2] [ 3  1  4][y] = [  1 -1/2   1/2][ 2]
{  8 -9/2   5/2] [-1  5 –2][z]   {  8 -9/2   5/2][-2]

Do the matrix multiplication A-1AX = A-1B becomes
                                IX = A-1B which becomes:
                                 X = A-1B
                                 like this:

                       I     X  = A-1B  

                   [1  0  0][x]   [-35]
                   [0  1  0][y] = [  3]
                   [0  0  1][z]   [ 26]

                     
                             X  = A-1B  

                            [x]   [-35]
                            [y] = [  3]
                            [z]   [ 26]
         
Edwin