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

Solution:

Given, $A^{T} A = 3 I_{3}$

$\begin{array}{l} \Rightarrow [\begin{array}{rrr} 0 & 2 x & 2 x \\ 2 y & y & - y \\ 1 & - 1 & 1 \end{array}] [\begin{aligned} 0 & 2 y & 1 \\ 2 x & y & - 1 \\ 2 x & - y & 1 \end{aligned}] = 3 [\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] \\ \Rightarrow [\begin{array}{rrr} 8 x^{2} & 0 & 0 \\ 0 & 6 y^{2} & 0 \\ 0 & 0 & 3 \end{array}] = [\begin{array}{lll} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{array}] \Rightarrow 8 x^{2} = 3, 6 y^{2} = 3 \\ \Rightarrow x^{2} = \frac{3}{8}, y^{2} = \frac{1}{2} \Rightarrow x = \pm \sqrt{\frac{3}{8}}, y = \pm \sqrt{\frac{1}{2}} \end{array}$

$∴$ Required number of matrices is 4.

The total number of matrices The total number of matrices $A = (\begin{array}{l} 0 2 y 1 \\ 2 x y - 1 \\ 2 x - y 1 \end{array})$ $(x, y \in R, x \neq y)$ for which $A^{T} A = 3 I_{3}$ is

Solution:

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