By Edward Gorey

A suite of the Edward Gorey`s haunting and surreal photograph tales and verses: The Unstrung Harp, The directory Attic, The uncertain visitor, The Object-Lesson, The malicious program ebook, The deadly Lozenge, The Hapless Shild, The Curious couch, The Willowdale Handcar, The Gashlycrumb Tinies, The Insect God, The West Wing, The Wuggly Ump, The Sinking Spell, The remembered stopover at.

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**Example text**

C) If A is a symmetric matrix, then so is An for each nEZ+. (d) If A, B are n x n symmetric matrices, then so is AB. 8 Let A be a square matrix, and suppose that A = B + C where B is symmetric and C is antisymmetric. Show that B=1{A +AT), c =1

Find An for all positive integers n. 4 PROPERTIES OF MATRIX MULTIPLICATION Let A, B, C be any real or complex matrices, and let a be any real or complex number. When all the following sums and products are defined, matrix multiplication satisfies the following properties: Mt (AB)C = A(BC). M2 A(B +C)= AB +A C. M3 (A+B)C=AC+BC . M4 a(AB) = (aA)B = A(aB). Implicit in all of these is that if either side is defined then so is the other, and equality then results. Property Ml is known as the associative law of multiplication; M2 and M3 are usually called the distributive laws of multiplication over addition.

Is there a practical criterion for determining whether a given square matrix is invertible? We will address these questions in the next two chapters. 2 Let A be ann x n matrix such that A 2 + 2A + I and find its inverse. = 0. Prove that A is invertible Solution We employ the strategy used in establishing properties I1 to 14 above. The equation satisfied by A can be rewritten as -A 2 -2A=l. But then, factorising the left-hand side yields A(- A - 21) = (-A - 2/)A = I. From this it is clear that A is invertible with inverse A - l = -A- 21.