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Laplace s Expansion Theorem The minor, Mij , of the element aij of a determinant of order n is the determinant of order n 1 obtained by deleting the row and column containing aij . The cofactor, ij , of the element aij is de ned as ij 1 i j Mij Laplace s theorem states: In the determinant of a square matrix A, multiply each element in the pth row (column) by the cofactor of the corresponding element in the qth row (column), and sum the products. Then the result is 0, for p 6 q; and det A, for p q. It follows at once from Laplace s theorem that if A has two rows or two columns the same, then det A 0 (and A must be a singular matrix).

Thus if /1 and /2 in Fig. 1-46 are both congruent and supplementary, then each of them is a right angle.

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Matrix Inversion by Determinants; Cramer s rule Laplace s expansion theorem can be exhibited as a matrix multiplication, as follows: 2 a11 a12 a22 ... an2 a13 a23 ... ... a1n 32 11 21 22 31 32 ... ... n1 3 6a 6 21 6 4 ... an1 2 a2n 76 12 76 76 . . . 54 . . . n2 7 7 7 ... 5

1.13 Naming pairs of angles (a) In Fig. 1-47(a), name two pairs of supplementary angles. (b) In Fig. 1-47(b), name two pairs of complementary angles. (c) In Fig. 1-47(c), name two pairs of vertical angles.

R1 0

11 6 6 12 6 4 ... 2

Fig. 1-47

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... ... ... ... ... 1n 2n 3n . . . nn an3 . . . ann 32 3 21 31 . . . n1 a11 a12 a13 . . . a1n 22 32 . . . n2 76 a21 a22 a23 . . . a2n 7 76 7 76 7 . . . . . . . . . . . . 54 . . . . . . . . . . . . . . . 5 an1 3 7 7 7 5 an2 an3 . . . ann

Find the input impedance Z1 of the inverting ampli er of Fig. 9-2, assuming the basic op amp is ideal.

(a) Since their sum is 180 , the supplementary angles are (1) /1 and /BED; (2) /3 and /AEC. (b) Since their sum is 90 , the complementary angles are (1) /4 and /FJH; (2) /6 and /EJG. (c) Since KL and MN are intersecting lines, the vertical angles are (1) /8 and /10; (2) /9 and /MOK.

where adj A ji is the transposed matrix of the cofactors of the aij in the determinant of A, and I is the n n unit matrix. If A is nonsingular, one may divide through by det A 6 0, and infer that A 1 1 adj A det A

1.14 Finding pairs of angles Find two angles such that: (a) The angles are supplementary and the larger is twice the smaller. (b) The angles are complementary and the larger is 20 more than the smaller. (c) The angles are adjacent and form an angle of 120 . The larger is 20 less than three times the smaller. (d) The angles are vertical and complementary.

Consider vS a driving-point source. Since the op amp is ideal, the inverting terminal is a virtual ground, and a loop equation at the input leads to vS i1 R1 0 so that Z1 vS R1 i1

This means that the unique solution of the linear system Y AX is 1 adj A Y X det A which is Cramer s rule in matrix form. The ordinary, determinant form is obtained by considering the rth row r 1; 2; . . . ; n of the matrix solution. Since the rth row of adj A is 1r 2r 3r . . . nr we have:

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