Home
Class 12
MATHS
If A = [(1,2,2),(2,1,-2),(a,2,b)] is a m...

If A = `[(1,2,2),(2,1,-2),(a,2,b)]` is a matrix satisfying the equation `"AA"^(T) = 9I`, where `I is 3xx3` identity matrix, then the ordered pair (a,b) is equal to

A

(2, -1)

B

(-2, 1)

C

(2, 1)

D

(-2, -1)

Text Solution

Verified by Experts

The correct Answer is:
D
Given, `A = [{:(1, 2, 2), (2, 1, -2), (a,2, b):}], A^(T) = [{:(1, 2, a), (2, 1, 2), (2, -2, b):}]` and
`"AA"^(T) = [{:(1, 2, 2), (2, 1, -2), (a,2, b):}] [{:(1, 2, a), (2, 1, 2), (2, -2, b):}]`
`= [{:(9, 0, a+ 4 +2b), (0, 9, 2a+2-2b), (a+4+2b, 2a+2-2b, a^(2) + 4 +b^(2)):}]`
It is given that, `"AA"^(T) = 9I`
`rArr [{:(9, 0, a+ 4 +2b), (0, 9, 2a+2-2b), (a+4+2b, 2a+2-2b, a^(2) + 4 +b^(2)):}] =9[{:(1, 0, 0), (0, 1, 0), (0, 0, 1):}]`
`rArr [{:(9, 0, a+ 4 +2b), (0, 9, 2a+2-2b), (a+4+2b, 2a+2-2b, a^(2) + 4 +b^(2)):}] =[{:(9, 0, 0), (0, 9, 0), (0, 0, 9):}]`
On comparing, we get
`a +4 +2b = 0 rArr a + 2b = -4 " " ...(i)`
`2a + 2 -2b =0 rArr a-b =-1 " " .... (ii)`
and `a^(2) + 4 + b^(2) = 9 " " ...(iii)`
On solving Eqs. (i) and (ii), we get
a= -2, b =-1
This satisfies Eq. (iii)
Hence, `(a, b) -= (-2, -1)`
Promotional Banner

Similar Questions

Explore conceptually related problems

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.

A square matrix P satisfies P^(2)=I-P where I is identity matrix. If P^(n)=5I-8P , then n is

A square matrix M of order 3 satisfies M^(2)=I-M , where I is an identity matrix of order 3. If M^(n)=5I-8M , then n is equal to _______.

If A is an invertible matrix of order 2, then det (A^(-1)) is equal to

If A is a skew symmetric matrix, then B=(I-A)(I+A)^(-1) is (where I is an identity matrix of same order as of A )

If M is a 3xx3 matrix, where det M=1a n dM M^T=1,w h e r eI is an identity matrix, prove theat det (M-I)=0.

If (3)/(2)+(7)/(2)i is a solution of the equation ax^(2)-6x+b=0 , where a and b are real numbers, then the value of a+b is equal to

If A is an idempotent matrix satisfying (I-0.4A)^(-1)=I-alphaA where I is the unit matrix of the same order as that of A then the value of alpha is

A skew-symmetric matrix A satisfies the relation A^2+I=O ,w h e r e I is a unit matrix then A is a. idempotent b. orthogonal c. of even order d. odd order