SOLUTION: Solve the following systems using inverse matrices. can you also show me the steps.. thanks 2x - y = 2 2x - 2y = 4 2x + 3y + 4z = 3 -13x + 5y - 2z = 3

Algebra ->  Algebra  -> College  -> Linear Algebra -> SOLUTION: Solve the following systems using inverse matrices. can you also show me the steps.. thanks 2x - y = 2 2x - 2y = 4 2x + 3y + 4z = 3 -13x + 5y - 2z = 3      Log On


   

Question 176635: Solve the following systems using inverse matrices. can you also show me the steps.. thanks


2x - y = 2
2x - 2y = 4









2x + 3y + 4z = 3
-13x + 5y - 2z = 3
-3x + 4y + 3z = 6

Answer by Mathtut(3670) About Me  (Show Source):
You can put this solution on YOUR website!
2x - y = 2
2x - 2y = 4
:
matrix A =%28matrix%282%2C2%2C2%2C-1%2C2%2C-2%29%29
matrix C =%28matrix%282%2C1%2C2%2C4%29%29
adj A=%28matrix%282%2C2%2C-2%2C1%2C-2%2C2%29%29 the adjunct of a 2x2 matrix is the matrix in which the main diagonal elements are switched and the other elements are multiplied by -1.
:
the inverse of A or A%5E-1 is
:
so det A=-2(2)-2(-1)=-2--->-1/2
A%5E-1=(1/det A)(adj A)=-1/2%28matrix%282%2C2%2C-2%2C1%2C-2%2C2%29%29=%28matrix%282%2C2%2C1%2C-1%2F2%2C1%2C-1%29%29
:
now finally the answer is found by multiplying the inverse of A by matrix CA%5E-1mat C
:
%28matrix%282%2C2%2C1%2C-1%2F2%2C1%2C-1%29%29%28matrix%282%2C1%2C2%2C4%29%29=%28matrix%282%2C1%2C0%2C-2%29%29
:
so x=0 and y=-2
::
::
::
::
2x + 3y + 4z = 3
-13x + 5y - 2z = 3
-3x + 4y + 3z = 6
:
RHS is the numbers on the right hand side of the equation.
C1 is column 1...etc
R1 is row 1...etc
:Matrix A %28matrix%283%2C3%2C2%2C3%2C4%2C-13%2C5%2C-2%2C-3%2C4%2C3%29%29
:
det A is the determinant of A
:
det A=(2) det%28matrix%282%2C2%2C5%2C-2%2C4%2C3%29%29-(-13)det%28matrix%282%2C2%2C3%2C4%2C4%2C3%29%29+(-3)det%28matrix%282%2C2%2C3%2C4%2C5%2C-2%29%29=46-91+78=33
:
det A=13
:
matrix A(C1 replaced with RHS) lets call A1
A1=%28matrix%283%2C3%2C3%2C3%2C4%2C3%2C5%2C-2%2C6%2C4%2C3%29%29
det A1=b>(3) det%28matrix%282%2C2%2C5%2C-2%2C4%2C3%29%29-(3)det%28matrix%282%2C2%2C3%2C4%2C4%2C3%29%29+(6)det%28matrix%282%2C2%2C3%2C4%2C5%2C-2%29%29=69+21-156=-66
:
matrix A(C2 replaced with RHS) lets call A2
A2=%28matrix%283%2C3%2C2%2C3%2C4%2C-13%2C3%2C-2%2C-3%2C6%2C3%29%29
det A2=b>(2) det%28matrix%282%2C2%2C3%2C-2%2C6%2C3%29%29-(-13)det%28matrix%282%2C2%2C3%2C4%2C6%2C3%29%29+(-3)det%28matrix%282%2C2%2C3%2C4%2C3%2C-2%29%29=42-195+54=-99
:
matrix A(C3 replaced with RHS) lets call A3
A3=%28matrix%283%2C3%2C2%2C3%2C3%2C-13%2C5%2C3%2C-3%2C4%2C6%29%29
det A3=b>(2) det%28matrix%282%2C2%2C5%2C3%2C4%2C6%29%29-(-13)det%28matrix%282%2C2%2C3%2C3%2C4%2C6%29%29+(-3)det%28matrix%282%2C2%2C3%2C3%2C5%2C3%29%29=36+78+18=132
:
x=det+A1%2Fdet+A=-66%2F33=highlight%28-2%29
x=det+A2%2Fdet+A=-99%2F33=highlight%28-3%29
x=det+A3%2Fdet+A=132%2F33=highlight%284%29