If $\mathrm{A}=\left[\begin{array}{ccc}1 & 2 & 2 \\ 2 & 1 & -2 \\ \mathrm{a} & 2 & \mathrm{~b}\end{array}\right]$ is a matrix satisfying the equation $\mathrm{AA}^{\mathrm{T}}=9 \mathrm{I}$, where I is $3 \times 3$ identity matrix, then ordered pair $(\mathrm{a}, \mathrm{b})$ is equal to :

JEE Main · 2015 · 04 Apr 2015 · Offline · Matrices

A = 12 a 212 2 - 2 b

is a matrix satisfying the equation

AA^{T} = 9 I

, where I is

3 \times 3

identity matrix, then ordered pair

(a, b)

is equal to :

Answer: (C)

The condition

A A^{T} = 9 I

says the rows of

A

are mutually orthogonal, each of squared length

9

. Orthogonality of the third row with the first gives

a + 4 + 2 b = 0

, and with the second gives

2 a + 2 - 2 b = 0

; these read

a + 2 b = - 4

and

a - b = - 1

. Subtracting,

3 b = - 3

b = - 1

, and then

a = - 2

. The ordered pair is

(- 2, - 1)

JEE Main · 2015 · 04 Apr 2015 · Offline · Single correct option