Find the values of x, y, z if the matrix satisfy the equation A'A = I3.

Engineering

Mathematics

Multiplication of Matrices

Symmetric and Skew Symmetric Matrices

Question

Find the values of x, y, z if the matrix $A = [\begin{array}{c} 0 \\ x \\ x \end{array} \begin{array}{c} 2 y \\ y \\ - y \end{array} \begin{array}{c} z \\ - z \\ z \end{array}]$ satisfy the equation A'A = I₃.

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$A = [\begin{array}{c} 0 \\ x \\ x \end{array} \begin{array}{c} 2 y \\ y \\ - y \end{array} \begin{array}{c} z \\ - z \\ z \end{array}]$

Given that A'A = I₃ ........(1)

$A' = [\begin{array}{ccc} 0 & x & x \\ 2 y & y & - y \\ z & - z & z \end{array}]$

$(1) \Rightarrow [\begin{array}{ccc} 0 & x & x \\ 2 y & y & - y \\ z & - z & z \end{array}] [\begin{array}{ccc} 0 & 2 y & z \\ x & y & - z \\ x & - y & z \end{array}] = [\begin{array}{lll} 1 0 0 \\ 0 1 0 \\ 0 0 1 \end{array}]$

$= [\begin{array}{ccc} 2 x^{2} & 0 & 0 \\ 0 & 4 y^{2} + 2 y^{2} & 0 \\ 0 & 0 & 3 z^{2} \end{array}] = [\begin{array}{lll} 1 0 0 \\ 0 1 0 \\ 0 0 1 \end{array}]$

$\begin{aligned} \Rightarrow 2 x^{2} = 1 \Rightarrow x \Rightarrow= \pm \sqrt{1 / 2} . \end{aligned}$

$\Rightarrow 6 y^{2} = 1 \Rightarrow y = \pm \sqrt{1 / 6}$

$\Rightarrow \begin{aligned} 3 z^{2} = 1 \Rightarrow z = \pm \sqrt{1 / 3} \end{aligned}$

$∴ x = \pm \sqrt{1 / 2}, y = \pm \sqrt{1 / 6}, z = \pm \sqrt{1 / 3}$

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