Question

x − y + z = 3
2x + y − z = 2
− x − 2y + 2z = 1

Answer 1

x − y + z = 3
2x + y − z = 2
− x − 2y + 2z = 1

This system can be written as AX = B,

`A = [(1,1,1),(2,-1,1),(1,-2,3)]`

`x = [(x),(y),(z)]`

`B = [(3),(2),(2)]`

|A| = 1(-3+2) –1 (6 - 1) + 1(-4 + 1)

= -1 - 5 - 3

= -9

`A_11 = 1 |(-1,1),(-2,3)| = -1`

`A_12 = -1|(2,1),(1,3)| = -5`

`A_13 = 1|(2,-1),(1,-2)| = -3`

`A_21 = -1 |(1,1),(-2,3)| = -5`

`A_22 = 1|(1,1),(1,3)| = 2`

`A_23 = -1 |(1,1),(1,-2)| = 3`

Hence adj A = `[(A_11,A_21,A_31),(A_12,A_22,A_32),(A_13,A_23,A_33)]`

`=[(-1,-5,2),(-5,2,1),(-3,3,-3)]`

`A^-1 = 1/|A| adjA`

`=1/-9[(-1,-5,2),(-5,2,1),(-3,3,-3)]`

`X = A^-1B`

`X = 1/-9 [(-1,-5,2),(-5,2,1),(-3,3,-3)][(3),(2),(2)]`

`[(x),(y),(z)] = - (-1)/9 [(-3-10+4),(-15+4+2),(-9+6-6)]`

`[(x),(y),(z)] = [((-9)/-9),((-9)/-9),((-9)/-9)] = [(1),(1),(1)]`

x = 1, y = 1, z = 1