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If A = \( \begin{bmatrix} cosa& sina \\[0.3em] -sina& cosa \\[0.3em] \end{bmatrix}\), show that A’A = I.

A = [(cosa, sina)(-sina, cosa)]

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Given A = \( \begin{bmatrix} cosa& sina \\[0.3em] -sina& cosa \\[0.3em] \end{bmatrix}\), We will find A’

\(\Rightarrow\) \( \begin{bmatrix} 1& 0 \\[0.3em] 0& 1 \\[0.3em] \end{bmatrix}\) [Using cos2a + sin2a = 1 and commutative law a.b = b.a i.e. sina cosa = cosa sina]

RHS = I \(\Rightarrow\) \( \begin{bmatrix} 1& 0 \\[0.3em] 0& 1 \\[0.3em] \end{bmatrix}\)

LHS = RHS 

Hence proved.

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