Question

Explain all parts of question 1 and question 2 in detail

1. Consider the matrix In + Inn, which has every diagonal entry equal to 2 and every off-diagonal entry equal to 1. (a) Compute det(In + Inn) for each of n = 1,2,3. (b) For n = 4, we have 2 1 1 1 1 2 1 1 1 1 2 1 111 2 2 1 1 1 -1 1 0 0 -1 0 1 0 -1 0 0 1 5 1 1 1 0 1 0 0 0 0 1 0 0 0 0 1 so the determinant equals 5 in this case. Justify that the determinants above are equal using row operations or column operations. (c) Conjecture the value of det(1, + Inn) for general n. 2. A square matrix A is skew-symmetric if AT = -A. (a) If A is an n x n skew-symmetric matrix, with n odd, prove that A is singular, i.e. non-invertible. (b) Find a skew-symmetric matrix that is invertible.

Answer 1

Solution : 2 1 1. ----11 : Here, Int Jual 1 2 1 - - 1 1 1 i 1 2 ---41 -- - - - - 1 1 - -_21 li 1.-.1 2 / ie. every diagonal entry equal to and every off-diagonal entry equal to nxn For n=1 det (I, +,,) = 121 = 2 For n=2, I det (I2 + J22) = 1 12 1 Ain 3 For no3 :. det (13+133) 2 . 52ľ -111/+11.11 = 2 (4-1) -1 (2-1) + (1-2) z 6-1-1 = 4.

1. € For n =4, we have - det (I + Ja4) = 2 - . I l - N 1.2 1 1 1 [ R2 = R₂R, Rs = R3-R, Ru= R4=R] [c'=C+ C+ Gtly] 1 2 I i 1 2 i 1 1 II 2. 1 1 2 1-1 1 1 - oo - ..oo - S XIXIXI 35

1.© det (Int Inn) 2 1 1 2 - - - II i i lum--21 --I 2 Inxn 2 1 1 - - 1 1 1 - 1 1 o. oo [ Ro'z R2 Ry o o R = R₃-R, Roya RaiR, - oo iö Ria Rar R7 -1 0 0 - --- o n xn n+1 0 1 1 ------| 1 0 - ---OO Tre • G+ + Czt & Cn- tch 7 O oo -- nxn This is an determinant upper triangular matrix, so, the is equal to product of diagonal elements. det (In + Inx) = +1) XIXIX--X1X1 =htla for general n.

2. A square if matrix A is skew.symmetrie ATS-A F is Definition: said to be Singular A square matrix A over a field hon singular if det A to and if det A=0 field F is Theorem 1. invertible An non matrix A over a iff A is non-singular ☆ Here, A is an nan skew-symmetric matrix, with n odd. deta = det AT - Therefore, AT = -A i det (AT) = det (-A) det A = (in de ta [ & A is a non . matrix] =>2deta=0 de tA=0 =) A is singular [ by difinition) =) A is non invertible [by theorem) a det A = - deta [in is add, som ③ Let A = (i ) AT= A is skew.symmetric matrix det A = 1361 - 170 A is non-singular A is invertible. O