Advanced Calculus fi..

Chapter 1 Victors and Matrices 31b) If n = 3, then A - - is the matrix1-det AC) Let Aij denote the minor determinant of A obtained by deleting the ith row and jthcolumn from A. Let hi, = (-]);+'A ji. The matrix B = (bij) is called the adjoint ofA and is denoted by adj A. Show that- 1A-I - adj A.det A8. Let all matrices occurring in the following equations be square of order n. Solve for Xand Y, stating which matrices are assumed to be nonsingular:9. a) Prove the rule (1.60) for 2 x 2 matrices; that is, prove:[Hint: Use Rule V of Section 1.4 for adding determinants to write the right-hand sideas a sum of four determinants. Then factor bl 1. . . . from these and show that twoterms are 0 and the other two have sumb) Prove the rule (1.60) for 3 x 3 matrices.C) Prove the rule ( 1.60) for n x n matrices.10. Let A = col (U 1, . . . . u,,) (column vector) and B = (vl , . . . , v,) (row vector); let n 2 2.a) Show directly that thc n x n matrix AB is singular (cf. the Remark at the end ofSection 1.9).b) Show that BA is 1 x I- that is, a scalar.11. (Permutation matrices) Let Pij be the n x n matrix obtained from I by interchangingthe ith and jth rows (i < j). Thus for n = 3,a) Show that P; = I.b) Show that Pij can be obtained from I by interchanging the ith and jth columns of I.c) Show that, if A is an n x n matrix, then PjjA is obtained from A by interchangingthe ith and jth rows of A, and A Pij is obtained from A by interchanging the ith andj th columns of A.d) Show that Pij is nonsingular.