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)))
