6.3 Geometrical Interpretations of the Determinant; Cramer’s Rule 277{ Hint:Consider the productIn0- C A C 62. Consider n x nmatrices A, B ,C,and Dsuch that rank(A) = rankShow thatA C D= a. D= CA1B,anddet(A) det (B)det (C) det(D) ible. [Hint:Consider the productb. The 2 x 2 matrixis noninvert-InO'A B'-C A" 1InC 63.Show that more than n\ =1 - 2 - 3.........nmultipli cations are required to compute the determinant of an n x nmatrix by Laplace expansion (for n> 2).64.Show that fewer than e• n! algebraic operations (addi tions and multiplications) are required to compute the determinant of an n x nmatrix by Laplace expansion. Hint:Let Lnbe the number of operations required to compute the determinant of a “general” n x nmatrix by Laplace expansion. Find a formula expressing Lnin terms of j. Use this formula to show, by induction, thatLn11- 4 = 1 + 1 + rr + XT + • •n\213!■+11( n -1)! n\Use the Taylor series of ex, ex= Y^n=onT ’t0 s^ow l^at the right-hand side of this equation is less than e.65. Let Mnbe the n x nmatrix with 1’s on the main diago nal and directly above the main diagonal, —1’s directlyMa=below the main diagonal, and 0’s elsewhere. For example," 1 10 -11100 - 1 1 1 0 0—1Letdn= det(M„).a.For n> 3, find a formula expressing dnin terms of dn-\and J„_2.b. Find d\, d2,^3, ^4, and d\Q.c. For which positive integers nis the matrix invertible?66. Let Mnbe the matrix with all l ’s along the main diago nal, directly above the main diagonal, and directly below the diagonal, and 0’s everywhere else. For example,^1 1 0 111001 1 10 0 1 1A/4 =Let dn= det(M„).a.Find a formula expressing dnin terms of dn- \and dn-2, for positive integers n >3.b. Find d \, d2,..., c.What is the relationship between dnand dn+3? What about dnand dn+6d. Find Jioo-67.Consider a pattern Pin an n x nmatrix, and choose an entry a{jin this pattern. Show that the number of inver sions involving a\jis even if (/ + 7) is even and odd if (/ + j)is odd. Hint:Suppose there are kentries in the pattern to the left and above atj. Express the number of inversions involving aijin terms of 68. Using the terminology introduced in the proof of The orem 6.2.10, show that sgnP = ( - l ) '+7sgn(P/y). See Exercise 67.Geometrical Interpretations of the Determinant; Cramer’s RuleWe now present several ways to think about the determinant in geometrical terms. Here is a preliminary exercise.ABDBn.D0“ 1Mn0^d%.?k.

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