Question

If A = `[(1, 2, 2),(2, 1, -2),(x, 2, y)]` is a matrix such that AA^T = 9I, find the values of x and y

Answer 1

A = `[(1, 2, 2),(2, 1, -2),(x, 2, y)]`

A^T = `[(1, 2, x),(2, 1, 2),(2, -2, y)]`

Given AA^T = 9I

`[(1, 2, 2),(2, 1, -2),(x, 2, y)] [(1, 2, x),(2, 1, 2),(2, -2, y)] = 9[(1, 0, 0),(0, 1, 0),(0, 0, 1)]`

`[(1 + 4 + 4, 2 + 2 - 4, x + 4 + 2y),(2 + 2 - 4, 4 + 1 + 4, 2x + 2 - 2y),(x + 4 + 24, 2x + 2 - 2y, x^2 + 4 + y^2)] = [(9, 0, 0),(0, 9, 0),(0, 0, 9)]`

Equating the corresponding entries
x + 4 + 2y = 0   .......(1)

2x + 2 – 2y = o .......(2)

x + 4 + 2y = 0   .......(3)

2x + 2 – 2y = 0  .......(4)

x² + 4 + y² = 9   .......(5)

(1) ⇒ x + 2y + 4 = 0
(2) ⇒ x –   y +  1 = 0
– ing      3y  +  3 = 0

y = `(-3)/3`

= – 1

Substituting the value of y in equation (1) we have

x + 4 + 2x – 1 = 0

x + 4 – 2 = 0

⇒ x = – 2

Substituting x = – 2 and y = – 1 in equation (5) we have

(5) ⇒ (– 2)² + 4 + (– 1)² = 9

4 + 4 + 1 = 9

9 = 9

∴ The required values of x and y are

x = – 2 and y = – 1