Solve system of linear equations, using matrix method, `x y" "+" "2z" "=" 7"` `3x" "+" "4y" "" "5z" "=" "" "5` `2x y" "+" "3" "z=" "12`

Explanation

`A=[{:(3,2),(1,1):}],`
`therefore" "A^(2)=A.A=[{:(3,2),(1,1):}][{:(3,2),(1,1):}]`
`[{:(9+2,6+2),(3+1,2+1):}]=[{:(11,8),(4,3):}]`
`"given that, "A^(2)+aA+bI=O`
`rArr" "[{:(11,8),(4,3):}]+a[{:(3,2),(1,1):}]+b[{:(1,0),(0,1):}]=0`
`rArr" "[{:(11+3a+b,8+2a),(4+a,3+a+b):}]=[{:(0,0),(0,0):}]`
`"On comparing,"4+a=0 rArr a=-4`
3+a+b=0
`rArr" "3-4+b=0`
`rArr" "b=1`
`therefore" "a=-4, b=1`