We start by calculating the determinant of matrix #A# to see if it's invertible
#det A=|(-3,-3,-4),(0,1,1),(4,3,4)|#
#=-3*|(1,1),(3,4)|+3*|(0,1),(4,4)|-4*|(0,1),(4,3)|#
#=(-3*1)+(3*-4)-4(-4)#
#=-3-12+16=1#
Therefore,
#detA!=0#, the matrix is invertible
We start by calculating the matrix of cofactors
#C=((|(1,1),(3,4)|,-|(0,1),(4,4)|,|(0,1),(4,4)|),(-|(-3,-4),(3,4)|,|(-3,-4),(4,4)|, -|(-3,-3),(4,3)|),(|(-3,-4),(1,1)|,-|(-3,-4),(0,1)|,|(-3,-3),(0,1)|))#
#C=((1,4,-4),(0,4,-3),(1,3,-3))#
Now, we determine the transpose of #C#
#C^T=((1,0,1),(4,4,3),(-4,-3,-3))#
So,
#A^-1=C^T/(detA)#
#A^-1=C=((1,0,1),(4,4,3),(-4,-3,-3))#
Verification
#A*A^-1=((-3,-3,-4),(0,1,1),(4,3,4))*((1,0,1),(4,4,3),(-4,-3,-3))#
#=((1,0,0),(0,1,0),(0,0,1))=I#
