New PDF release: Analysis of Structural Member Systems

By J J Connor

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B1, b2, a2, . . , [b1, b2 . (b) {ai, (c) [c1 01 [a11 a12 [o c2j [a21 a22 . , b,j 3 PROBLEMS 35 (d) 1—3. [Cii a121 0 [021 022j [0 C2 Show that the product of Sl= a1+ a2 + 03 = S2 + b2 + b3 = bk = can be written as 3 3 S1S2 I 1 Generalize this result for the sum of n elements. 1—4. Suppose the elements of a and b are functions of y. Let cia — [dv j dy db — [dblk] dy — [dy j Using (1—19), show that if c = ab then dc dy 1—5. db dy da dy =a—+—b Consider the triple product, abc. When is this product defined?

0 0 bflfl introduce symmetrical partitions after row (and column) p and write the product as We (pxp) (pxp) (pxq) [A1, [G1, A121 A22J — [G2, LA21 (qxp) (qxp) (qxq) (pxq) (pxp) 0 1[Bi, G22J[0 (qxq) (qxp) (pxq) B12 B22 (qxq) Note that the diagonal submatrices of g and b are triangular in form. 11 G11B12 = A12 G21B11 = A21 + G22B22 = A22 G21B12 (b) Show that A111 = 1G111 BijI and al = 1G111 G22j 1B111 1B221 (c) Suppose we require that By taking p = 1, 2,. . ,n — 1, deduce that this requirement leads to the INTRODUCTION TO MATRIX ALGEBRA 40 CHAP.

Tn general, a and b are equivalent if b = paq (1—54) where p and q are nonsinqular. This follows from the fact that the elementary operation matrices are nonsingular. 1-12. RANK OF A MATRIX The rank, r, of a matrix is defined as the order of the largest square array, formed by deleting certain rows and columns, which has a nonvanishing deter- minant. The concept of rank is quite important since, as we shall see in the next section, the solvability of a set of linear algebraic equations is dependent on the rank of certain matrices associated with the set.

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Analysis of Structural Member Systems by J J Connor

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