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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. These procedures are described in References 9—13.

As • • + I when — 1 when . is . , an even permutation (1—43) . , a,,) is an odd permutation Using (1—43), the definition equation for an ,ith-order determinant can be written as a11 a12 a1,, a21 a22 a2,, = (1—44) 1 where the summation is taken over all possible permutations of (1, 2, Factorial n = = n(n — 1)(n — 2) . • (2)(1). . , n). INTRODUCTION TO MATRIX ALGEBRA 18 CHAP. 1 Example 1—8 The permutations for n = 3 are cxi—1 x23 a33 a32 =2 1 =3 a1=1 z1=2 a3=1 a32 a3=1 e123=+1 e132=—1 e231=+1 e312=+1 e321—-—1 Using (1—44), we obtain a11a22a33 — a11a23a32 a11 a12 a13 a21 a22 a23 = —a12a21a33 + a12a23a31 a32 a33 +a13a21a32 — a13a22a31 This result coincides with (1—42).

INTRODUCTION Consider the second-order homogeneous system, (ajj 2)x1 + at2xz 0 a25x1 + (a22 — A)x2 = 0 where A is a scalar. Using matrix notation, we can write (2—i) as ax (2—2) Ax or (a — 212)x 0 (2—3) The values of 2. for which nontrivial solutions of (2—i) exist are called the characteristic values of a. Also, the problem of finding the characteristic values and corresponding nontrivial solutions of (2—i) is referred to as a second-order characteristic-value problem,* problem occurs naturally in the free-vibration The analysis of a linear system.

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