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By Lucia M., Magtone R., Zhou H.-S.

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52 implies that A is nonsingular, and so B = A−1 O = O. 6. Never — its determinant is always zero. 7. 82, 83) and commutativity of numeric multiplication, 1 det A det S = det A. 8. Multiplying one row of A by c multiplies its determinant by c. To obtain c A, we must multiply all n rows by c, and hence the determinant is multiplied by c a total of n times. 9. 56, det LT = det L. If L is a lower triangular matrix, then LT is an 37 upper triangular matrix. 50, det LT is the product of its diagonal entries which are the same as the diagonal entries of L.

W1 (x, y) v1 (x, y) + w1 (x, y) w1 (x, y) v1 (x, y) + = = + w2 (x, y) v2 (x, y) + w2 (x, y) w2 (x, y) v2 (x, y) Associativity of Addition: ! " ! # = = " u1 (x, y) + v1 (x, y) + w1 (x, y) u2 (x, y) + v2 (x, y) + w2 (x, y) u1 (x, y) u2 (x, y) ! Additive Identity: 0 = (0, 0) for all x, y, and ! v1 (x, y) v1 (x, y) +0= =0+ v2 (x, y) v2 (x, y) Additive Inverse: − v1 (x, y) v2 (x, y) v1 (x, y) v2 (x, y) ! + ! = ! − v1 (x, y) , and − v2 (x, y) − v1 (x, y) − v2 (x, y) ! =0= 43 v1 (x, y) v2 (x, y) + v1 (x, y) v2 (x, y) v1 (x, y) v2 (x, y) − v1 (x, y) − v2 (x, y) !

Det B splits into two terms, and we find that det B = det A + c det C, where C is the matrix obtained from A by replacing row l by row k. But rows k and l of C are identical, and so, by axiom (ii), if we interchange the two rows det C = − det C = 0. Thus, det B = det A. (ii) Let B be obtained from A by interchanging rows k and l. Then each summand in the formula for det B equals minus the corresponding summand in the formula for det A, since the permutation has changed sign, and so det B = − det A.

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A dirichlet problem with asymptotically linear and changing sign nonlinearity by Lucia M., Magtone R., Zhou H.-S.

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