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Common sense circuits have gotten more and more liable to probabilistic habit as a result of exterior radiation and approach edition. moreover, inherently probabilistic quantum- and nano-technologies are at the horizon as we method the bounds of CMOS scaling. making sure the reliability of such circuits regardless of the probabilistic habit is a key problem in IC design---one that necessitates a basic, probabilistic reformulation of synthesis and checking out innovations.

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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).

Oii — %1)(a22 — — a12a21 = 0 Since only one equation isindcpendcnt and there are two unknowns, the soluAssuming* tiOn is not unique. We define as the solution for A = 0, the solution of the first equation is that a12 xi') = 1) C1 = 012 where c1 is an arbitrary constant. Continuing, we let and take c1 such that = 1. This operation is called normalization, and the resulting column matrix, denoted by Q1, is referred to as the characteristic vector for = = —_2112 + L By definition, if a12 (2—10) a12 Q1Q1 = 1 0, we work with the second equation.

Referring to Example 1 —12, the matrices 1 1/2 1/5 3 7 2 —21 5 and 1 1/2 0 1 00 1/5 14/55 1 are equivalent. 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.