Question

The sum of three numbers is 2. If twice of the second number is added to the sum of first and third number we get 0, Adding five times the first number to twice the sum of second and third number we get 7. Find the numbers using matrix method.

Answer 1

Let the numbers be x, y and z.

According to the given conditions

x + y + z = 2

x + 2y + z = 0

And 5x + 2y + 2z = 7

These equations can be written in matrix form as:

`[(1, 1, 1),(1, 2, 1),(5, 2, 2)][(x),(y),(z)] = [(2),(0),(7)]`

By R₂ – R₁ and R₃ – 5R₁, we get

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

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

By equality of matrices, we get

x + y + z = 2   ...(1)

y = –2

–3y – 3z = –3   ...(2)

Substituting y = –2 in (2), we get

–3(–2) – 3z = –3

∴ –3z = –3 – 6 = –9

∴ –3z = –9

∴ z = 3

Substituting y = –2, z = 3 in (1), we get

x – 2 + 3 = 2

∴ x = 1

∴ x = 1, y = –2, z = 3.

Hence, the required numbers are 1, –2 and 3.