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.