Solve the following equations by matrix method. `2x + y + z = 1, x - 2y - 3z=1 and 3x + 2y +4z=5`

Explanation

Write the given equations in the form of matrices.
`[{:(2,1,1),(1,-2,-3),(3,2,4):}][{:(x),(y),(z):}]=[{:(1),(1),(5):}]`
`rArr" "AX=B`
`|A|=|{:(2,1,1),(1,-2,-3),(3,-2,4):}|`
`=2(-8+6)-1(4+9)+1(2+6)`
`=-4-13+8=-9ne0`
`therefore" A is invertible"`
Cofactors of the elements of matrex A.
`c_(11)=(-1)^(1+1)|{:(-2,-3),(2,4):}|=-8+6=-2`
`c_(12)=(-1)^(1+2)|{:(1,-3),(3,4):}|=-(4+9)=-13`
`c_(13)=(-1)^(1+3)|{:(1,-2),(3,2):}|=2+6=8`
`c_(21)=(-1)^(2+1)|{:(1,1),(2,4):}|=-(4-2)=-2`
`c_(22)=(-1)^(2+2)|{:(2,1),(3,4):}|=8-3=5`
`c_(23)=(-1)^(2+3)|{:(2,1),(3,2):}|=-(4-3)=-1`
`c_(31)=(-1)^(3+1)|{:(1,1),(-2,-3):}|=-3+2=-1`
`c_(32)=(-1)^(3+2)|{:(2,1),(1,-3):}|=-(-6-1)=7`
`c_(33)=(-1)^(3+3)|{:(2,1),(1,-2):}|=-4-1=-5`
`therefore" "adj.A"=|{:(c_(11),c_(12),c_(13)),(c_(21),c_(22),c_(23)),(c_(31),c_(32),c_(33)):}|=[{:(-2,-13,8),(-2,5,-1),(-1,7,-5):}]`
`[{:(-2,-2,-1),(-13,5,7),(8,-1,-5):}]`
`"and "A^(-1)=1/(|A|)"adj.A"=-1/9[{:(-2,-2,-1),(-13,5,7),(8,-1,-5):}]`
`"Now " AX=B`
`rArr" "X=A^(-1)B`
`rArr" "[{:(x),(y),(z):}]=-1/9[{:(-2,-2,-1),(-13,5,7),(8,-1,-5):}]-1/9[{:(1),(1),(5):}]`
`=-1/9[{:(-2,-2,-5),(-13,+5,+35),(8,-1,-25):}]`
`=-1/9[{:(-9),(27),(-18):}]=[{:(1),(-3),(2):}]`
`therefore" "x=1,y=-3,z=2`.