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The total number of matrices A = [{:(0,...

The total number of matrices `A = [{:(0, 2y, 1), (2x, y, -1), (2x, -y, ):}] (x, y in R, x ne y) " for which "A^(T)A = 3I_(3)` is

A

2

B

4

C

3

D

6

Text Solution

Verified by Experts

The correct Answer is:
D
Given matrix
` A= [{:(0, 2y, 1), (2x, y, -1), (2x, -y, 1):}], (x, y in R, x ne y)`
for which
`A^(T) A = 3I_(3)`
`rArr [{:(0, 2x, 2x), (2y, y, -y), (1, -1, 1):}][{:(0, 2y, 1), (2x, y, -1), (2x, -y, 1):}] = [{:(3, 0, 0), (0, 3, 0), (0, 0, 3):}]`
`rArr [{:(8x^(2), 0, 0), (0, 6y^(2), 0), (0, 0, 3):}] = [{:(3, 0, 0), (0, 3, 0), (0, 0, 3):}]`
Here, two matrices are equal, therefore equating the corresponding elements, we get
`8x^(2) = 3 " and " 6y^(2) = 3`
`rArr x = +- sqrt((3)/(8))`
`"and " y = +- (1)/(sqrt(2))`
`therefore` There are 2 different values of x and y each.
So, 4 matrices are possible such that `A^(T)A = 3I_(3)`.
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