Question

If A = `[(-3, -2, -4),(2, 1, 2),(2, 1, 3)]`, B = `[(1, 2, 0),(-2, -1, -2),(0, -1, 1)]` then find AB and use it to solve the following system of equations:

x – 2y = 3

2x – y – z = 2

–2y + z = 3

Answer 1

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

and B = `[(1, 2, 0),(-2, -1, -2),(0, -1, 1)]`

Now, AB = `[(-3, -2, -4),(2, 1, 2),(2, 1, 3)][(1, 2, 0),(-2, -1, -2),(0, -1, 1)]`

= `[(-3 + 4 + 0, -6 + 2 + 4, 0 + 4 - 4),(2 - 2 + 0, 4 - 1 - 2, 0 - 2 + 2),(2 - 2 + 0, 4 - 1 - 3, 0 - 2 + 3)]`

= `[(1, 0, 0),(0, 1, 0),(0, 0, 1)]`

∴ AB = I

(AB)B^–1 = IB^–1

A = B^–1  ...(i)

Now, equations are

x – 2y = 3

2x – y – z = 2

– 2y + z = 3

∴ `[(1, -2, 0),(2, -1, -1),(0, -2, 1)][(x),(y),(z)] = [(3),(2),(3)]`

CX = D

Here, C = B^T  ...(ii)

∴  C^–1(CX) = C^–1 D

`\implies` IX = C^–1 D 

`\implies` X = C^–1 D

= [B^T]^–1 D  ...[From (ii)]

= [B^–1]^T D 

= [A]^T D  ...[From (i)]

X = `[(-3, -2, -4),(2, 1, 2),(2, 1, 3)]^T[(3),(2),(3)]`

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

= `[(-9 + 4 + 6),(-6 + 2 + 3),(-12 + 4 + 9)]`

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

Hence x = 1, y = – 1 and z = 1.