Question

Find the product of the matrices `[(1, 2, -3),(2, 3, 2),(3, -3, -4)][(-6, 17, 13),(14, 5, -8),(-15, 9, -1)]` and hence solve the system of linear equations:

x + 2x − 3z = −4

2x + 3y + 2z = 2

3x − 3y − 4z = 11 

Answer 1

We have, 

`AB = [(1, 2, -3),(2, 3, 2),(3, -3, -4)][(-6, 17, 13),(14, 5, -8),(-15, 9, -1)]`

= `[(-6 + 28 + 45, 17 + 10 - 27, 13 - 16 + 3),(-12+42-30, 34 + 15 + 18, 26 - 24 - 2),(-18 - 42 + 60, 51 - 15 - 36, 39 + 24 + 4)]`

= `[(67, 0, 0),(0, 67, 0),(0, 0, 67)]`

= 67`[(1, 0, 0),(0, 1, 0),(0, 0, 1)]`

Thus, AB = 67 I

⇒ `A(1/67B) = I`

⇒ A⁻¹ = `1/67(B)`

Given system of linear equation is 

x + 2y − 3z = −4

2x + 3y + 2z = 2

3x − 3y − 4z = 11 

Represent it in matrix form as 

`[(1, 2, -3),(2, 3, 2),(3, -3, -4)][(x),(y),(z)] = [(-4),(2),(11)]`

which is of the form AX = D

∴ X = A⁻¹D 

= `1/67(B).D`

= `1/67[(-6, 17, 13),(14, 5, -8),(-15, 9, -1)][(-4),(2),(11)]`

= `1/67[(201),(-134),(67)]`

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

∴ x = 3, y = −2 and z = 1