By J J Connor
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Extra info for Analysis of Structural Member Systems
This result is quite useful. * See Probs. 1—20, 1—21. f See Ref. 4, sect. 3 15, for a discussion of the general Laplace expansion method. The expansion in terms of cofactors for a iow Or a COlUmn is a special case of the general method. CRAMER'S RULE SEC. 1—9. The evaluation of a determinant, using the definition equation (1—44) or the cofactor expansion formula (1—46) is quite tedious, particularly when the array is large. A number of alternate and more efficient numerical procedures for evaluating determinants have been developed.
1 2 1 3 3 x1 3 5 X2 7 ii x3 1—28. = 3 5 Determine the adjoint matrix for a= Does a 2 123 1 3 5 3 7 11 [1 3 [3 2 exist? T = Find the inverse of Show that b1' bT, (a) (b) [2 4 [i S (c) [1 [3 31[2 4 2J[l s (d) [2 O1[2 [o 3j[l s Let a12 a = 31 [A11 A12 = [A2, A22 33 32 and B32 — [8,, 82, where the order of BJk is the same as AJ1. Starting with the condition aa' 13 PROBLEMS 41 determine the four matrix equations relating BJk and Aft (j, k result to find the inverse of 1—3!. 1, 2). Use this 124 212 121 Find the inverse of A lq L0 Note that A is (p x q).
It can be shown* that the determinant of the product of two square matrices is equal to the product of the determinants: ci = (1—48) a! hi Whether we use (1—48) or first multiply a and b and then determine lab! depends on the form and order of a and b. If they are diagonal or triangular, (1—48) is quite efficient. t Example 1—10 [1 r2 31 5] a! = hi 3 4 = Ic! = —20 Alternatively, [[11 c and 29J [1 a=[0 —20 r2 0 b__[1 31 5] bi = 8 a! = 5 cj = Ic! = +40 Determining c first, we obtain rs 121 = [5 20] 1—9.