Question

Compute the determinant of the following n x n matrix: [ 2-1 -1 2-1 -1 2 2 -1 -1 2 (All "missing" entries are 0.) This is a nice exercise in mathematical induction. To do this problem, first try computing a few specific cases, then make a conjecture about the general nxn determinant. Then prove that your conjecture is correct by induction. (Actually, you will use strong induction, where you assume that the determinant in the nxn case follows from the knowing the determinants in all lower order cases. It's equivalent to standard induction.) Incidentally, this matrix arises in an area of mathematics known as representation theory.

Answer 1

Given 2 -1 trist of all consider a 2x2 -12) matrix Thon A=(2, 2) 191-2.2-1 4-1 23 if A is 3x3 malsin -107 4 -12-1 - 21 141 -2(3)+1[2] = 6-2 4. if A is uxų malsix . [& 1 0 0 0 A-412-lo lot 2-1 lo lo -1 a i det of 3x3 Matrix

12 - 0 1 -1 0 JA: 2-1 2-1 Hilo 2-1 O-L-2 Tol 2 Tourettin 42 :2[4]- 1(3) -3 =dx(de l of 3x3 malsief-1 (det of 242 matsire) by induction assume that det of Anun malsix is (141) ie A=12-1 - 2 - T 0 :-/ 2Jnxn TA : (n+1) Let A be a (mto)y (m41) matris The A: 1 3 :.. 1 0 -72 Jembulxlmdu)

= 2x (del of nxn maloin ) - 11 del of (m-1)x (n-1) malvive) 12-1. ·2-12.. 1-1 -10.... 0 0 2-1 0 + 1 -122 Inxn 2 det of (m-1x (n-1) matrina :21m4ll-lem) M42 Hence by mathematic induction et Anxon Tal-0041.