x-y+9z=-27 2x-4y-z=-1 3x+6y-3z=27 solving 3 equations with 3 unknowns by matrices

Question

Solving 3 equations by matrices

Answer 1

Input all the coefficients of the variables in its respective rows and columns inside the matrix you have on you calculator. Then find its determinant. after finding the determinant of the coefficients, you change the date you've input in the first column (which are all coefficients of 'x' variable) with the constants (or the answer after the equal sign), then you find its determinant.

The determinant of you got on your 2nd matrix divided by the determinant of the coefficients is equal to the value of X.

Do same with the rest variables (y and z).

Answer 2

Equation in matrix format is:

( 1  -1  9 ) ( x )    ( -27 )

( 2 -4  -1 ) ( y ) = ( -1  )

( 3   6 -3 ) ( z )     (  27 )
Call this matrix M.
The determinant for the matrix is 1*(12+6)+1*(-6+3)+9(12+12)=18-3+216=231.

The transposition matrix to adjugate matrix (step by step):

(  1  2   3 )    ( 12+6  3-54 1+36 )         ( 18  51  37 )

( -1 -4  6 ) = ( -6+3 -3-27 -1-18 )       = ( 3  -30  19 )

(  9 -1  -3 )     ( 12+12 6+3 -4+2 )        ( 24  -9   -2 )

Note that -51, -3, -19 and 9 take on reversed signs.

Divide by the determinant 231 to get the inverse matrix (M-1)

Multiply the inverse matrix on both sides of the original equation in matrix format:

( x )               ( 18  51  37 ) ( -27)               ( 462 )    (  2 )

( y ) = 1/231 (    3 -30 19 ) ( -1  )  = 1/231 ( 462 ) = (  2 )

( z )               ( 24  -9  -2 )  (  27 )               (-693 )    ( -3 )

So the solution using matrices is x=y=2, z=-3.