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<p><b>Exercise</b>. Prove that the set <i>T</i> of matrices of the form (<i>a, b; -b a</i>) where <i>a,b</i> ∈ ℝ are not both null at the same time, is a subgroup of (GL<sub><i>n</i></sub>(ℝ), ⋅). <h2>Solution</h2> <p>The product of elements of <i>T</i> is still an element of <i>T</i> : (<i>a<sup>2</sup> −b<sup>2</sup> 2ab; -2ab a<sup>2</sup> −b<sup>2</sup></i>). For <i>a</i> = 1, <i>b</i> = 0 we have <i>I</i>. The inverse of a matrix of the form(<i>a, b; −b a</i>) is</p> <p>(<i>a/(<i>a<sup>2</sup> + b<sup>2</sup></i>) −<i>b</i>/(<i>a<sup>2</sup> + b<sup>2</sup></i>); <i>b</i>/(<i>a<sup>2</sup> + b<sup>2</sup></i>) <i>a</i>/(<i>a<sup>2</sup> + b<sup>2</sup></i>)</p>
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Fri, 19 Mar 2021 10:36 GMT