The total number of matrices The total number of matrices A= 0       2y        12x      y     −12x    −y       1 (x,y∈R,x≠y) for which ATA=3I3 is

# The total number of matrices The total number of matrices  $\left(x,y\in \mathbit{R},x\ne y\right)$ for which ${A}^{T}A=3{I}_{3}$ is

1. A

4

2. B

3

3. C

2

4. D

6

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### Solution:

Given, ${A}^{T}A=3{I}_{3}$

$\begin{array}{l}⇒\left[\begin{array}{rrr}0& 2x& 2x\\ 2y& y& -y\\ 1& -1& 1\end{array}\right]\left[\begin{array}{rlr}0& 2y& 1\\ 2x& y& -1\\ 2x& -y& 1\end{array}\right]=3\left[\begin{array}{lll}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\\ ⇒\left[\begin{array}{rrr}8{x}^{2}& 0& 0\\ 0& 6{y}^{2}& 0\\ 0& 0& 3\end{array}\right]=\left[\begin{array}{lll}3& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\right]⇒8{x}^{2}=3,6{y}^{2}=3\\ ⇒{x}^{2}=\frac{3}{8},{y}^{2}=\frac{1}{2}⇒x=±\sqrt{\frac{3}{8}},y=±\sqrt{\frac{1}{2}}\end{array}$

$\therefore$ Required number of matrices is 4.

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