Question

Given `A = [(-4, 4, 4), (-7, 1, 3), (5, -3, -1)] and B = [(1, -1, 1), (1, -2, -2), (2, 1, 3)]`, find AB. Hence, solve the system of linear equations:

x − y + z = 4

x − 2y − 2z = 9

2x + y + 3z = 1

Answer 1

`A = [(-4, 4, 4), (-7, 1, 3), (5, -3, -1)] and B = [(1, -1, 1), (1, -2, -2), (2, 1, 3)]`

AB = `[(-4, 4, 4), (-7, 1, 3), (5, -3, -1)][(1, -1, 1), (1, -2, -2), (2, 1, 3)]`

= `[(-4 + 4 + 8, 4 - 8 + 4, -4 - 8 + 12), (-7 + 1 + 6, 7 - 2 + 3, -7 - 2 + 9), (5 - 3 - 2, -5 + 6 - 1, 5 + 6 - 3)]`

AB = `[(8, 0, 0), (0, 8, 0), (0, 0, 8)]`

= 8 I    ...(i)

Given: x − y + z = 4

x − 2y − 2z = 9

2x + y + 3z = 1

In matrix form

`[(1, -1, 1), (1, -2, -2), (2, 1, 3)][(x), (y), (z)] = [(4), (9), (1)]`

BX = C

X = B⁻¹C

|B| = 0

From equation (i)

AB = 8 I

ABB⁻¹ = 8 I B⁻¹

AI = 8 B⁻¹

`1/8 A = B^-1`

X = `1/8 AC`

= `1/8[(-4, 4, 4), (-7, 1, 3), (5, -3, -1)][(4), (9), (1)]`

`[(x), (y), (z)] = 1/8[(-16 + 36 + 4), (-28 + 9 + 3), (20 - 27 - 1)]`

= `1/8[(24), (-16), (-8)]`

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

Hence, x = 3, y = −2, z = −1