Deliberately not using the shortcut method of using the determinant.
Write as # AX=B#
Then #A^(-1) AX=A^(-1)B#
But # A^(-1)A=I#
#=> X=A^(-1)B#
# color(red)("Note that the order of the matrices is important")#
#color(magenta)(A^(-1) A!=A A^(-1))#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Determine "A^(-1))#
#((-3,-2" |" 1,0),(" "1, 1 color(white)(..)" |"0,1))#
'...............................................
#Row( 1) + 3Row(2)#
#((0,1" |" 1,3),(1, 1 " |"0,1))#
'....................................................
#Row(2)-Row(1)#
#((0,color(white)(...)1" |"color(white)(..) 1,3),(1,color(white)(..) 0 " |"-1,-2))#
'................................................
Reverse the order of the rows
#((1,color(white)(..) 0 " |"-1,-2),(0,color(white)(.)1" |"color(white)(..) 1,3))#
'.......................................................
#color(brown)(=>A^(-1)=((-1,-2),(1,3)))#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#=> X = A^(-1)B#
#=> X = ((-1,-2),(1,3))((-8,-1),(6,0))#
#"Let " X = ((a,b),(c,d))#
#a=[(-1)xx(-8)]" "+" "[(-2)xx(6) ]= -4#
#b=[(-1)xx(-1)]" "+" "[(-2)xx(0)] = +1#
#c=[(1)xx(-8)]" "+" "[(3)xx(6)] = +10#
#d=[(1)xx(-1)]" "+" "[(3)xx(0)]=-1#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(green)(=>X = ((-4,+1),(+10,-1)))#
