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Question:

The total number of matrices A = ⎛⎜⎝0 2x 2x 2y y -y 1 -1 1⎞⎟⎠, (x, y ∈ R, x ≠ y) for which ATA = 3I3 is:

6

4

2

3

Solution:

The correct option is D
4
ATA = 3I3
⎛⎜⎝0 2x 2x 2y y -y 1 -1 1⎞⎟⎠⎛⎜⎝0 2y 1 2x y -1 2x -y⎞⎟⎠ = ⎛⎜⎝3 0 0 0 3 0 0 0 3⎞⎟⎠
⎛⎜⎝8x2 0 0 0 6y2 0 0 0 3⎞⎟⎠ = ⎡⎢⎣3 0 0 0 3 0 0 0 3⎤⎥⎦
8x2 = 3; 6y2 = 3
⇒ x2 = 3/8; y2 = 1/2
⇒ x = ±√(3/8); y = ±√(1/2)
Thus total number of matrices can be formed with these values are 4