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Given \(k=\frac{1}{\sqrt{50}}\) and the condition \(PP^T=I\), where \[ P=\begin{bmatrix} \frac{2}{3} & 3k & a\\ -\frac{1}{3} & -4k & b\\ \frac{2}{3} & -5k & c \end{bmatrix}, \] what is the value of \(a\)?

1. a = \(\pm\frac{13}{2\sqrt5}\)
2. b = \(\pm\frac{16}{5\sqrt2}\)
3. c = \(\pm\frac{13}{5\sqrt2}\)
4. c = \(\pm\frac{1}{2\sqrt3}\)

Correct answer: a = \(\pm\frac{13}{2\sqrt5}\)

Solution

Since \(PP^T=I\), each row of \(P\) has norm 1 and different rows are orthogonal. Using the first row, \(\left(\frac23\right)^2+(3k)^2+a^2=1\) with \(k=\frac1{\sqrt{50}}\). This gives \(\frac49+\frac{9}{50}+a^2=1\), so \(a^2=\frac{169}{20}\), hence \(a=\pm\frac{13}{2\sqrt5}\).

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