Question

If A = `[(2, -3, 5),(3, 2, -4),(1, 1, -2)]`, find A^–1. Use A^–1 to solve the following system of equations 2x − 3y + 5z = 11, 3x + 2y – 4z = –5, x + y – 2z = –3

Answer 1

A = `[(2, -3, 5),(3, 2, -4),(1, 1, -2)]`

|A| = 2(0) + 3(–2) + 5(1) = –1

A^–1 = `(adj A)/|A|`

adj A = `[(0, -1, 2),(2, -9, 23),(1, -5, 13)]`, A^–1 = `1/((-1)) [(0, -1, 2),(2, -9, 23),(1, -5, 13)]`

X = A^–1B

⇒ `[(x),(y),(z)]`

= `1/((-1)) [(0, -1, 2),(2, -9, 23),(1, -5, 13)][(11),(-5),(-3)]`

= `1/((-1)) [(0 + 5 - 6),(22 + 45 - 69),(11 + 25 - 39)]`

⇒ `[(x),(y),(z)] = 1/((-1)) [(-1),(-2),(-3)]`

⇒ x = 1, y = 2, z = 3.