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s1131 - PROBLEM 31 The Matrix L is dened by L= l00 l10 l20...

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PROBLEM 31 The Matrix L is defined by: L = l 0 0 l 0 1 l 0 2 l 0 3 l 1 0 l 1 1 l 1 2 l 1 3 l 2 0 l 2 1 l 2 2 l 2 3 l 3 0 l 3 1 l 3 2 l 3 3 . (a)–Calculate - gL . (b)–Write down the transpose matrix L . (c)–Calculate Lg . (d)–Compare (a) and (c) to find the general form of L ( i.e. use Lg = - gL ). (a)–The matrix g is defined by: g = 1 0 0 0 0 - 1 0 0 0 0 - 1 0 0 0 0 - 1 . –By matrix multiplication: - gL = - l 0 0 - l 0 1 - l 0 2 - l 0 3 l 1 0 l 1 1 l 1 2 l 1 3 l 2 0 l 2 1 l 2 2 l 2 3 l 3 0 l 3 1 l 3 2 l 3 3 . (b)–The transpose of L is: L = l 0 0 l 1 0 l 2 0 l 3 0 l 0 1 l 1 1 l 2 1 l 3 1 l 0 2 l 1 2 l 2 2 l 3 2 l 0 3 l 1 3 l 2 3 l 3 3 . (c)–By matrix multiplication: Lg = l 0 0 - l 1 0 - l 2 0 - l 3 0 l 0 1 - l 1 1 - l 2 1 - l 3 1 l 0 2 - l 1 2 - l 2 2 - l 3 2 l 0 3 - l 1 3 - l 2 3 - l 3 3 .
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(d)–Set - gL = Lg and compare the components: - l 0 0 = l 0 0 - l 0 1 = - l 1 0 - l 0 2 = - l 2 0 - l 0 3 = - l 3 0 l 1 0 = l 0 1 l 1 1 = - l 1 1 l 1 2 = - l 2 1 l 1 3 = - l 3 1 l 2 0 = l 0 2 l 2 1 = - l 1 2 l 2 2 = - l 2 2 l 2 3 = - l 3 2 l 3 0 = l 0 3 l 3 1 = - l 1 3 l 3 2 = - l 2 3 l 3 3 = - l 3 3 . (1) –The resulting matrix is: L = 0 l 0 1 l 0 2 l 0 3 l 0 1 0 l 1 2 l 1 3 l 0 2 - l 1 2 0 l 2 3 l 0 3 - l 1 3 - l 2 3 0 . (2) (e)–Obtain the same result by discussing the elements of
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