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Question 83308: 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
(ii)Obtain the determinant and inverse of A.
(iii)Solve the system.
Answer by Edwin McCravy(20055)   (Show Source):
You can put this solution on YOUR website!
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
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