Question

6. Let A be a nxn matrix and let B be the matrix composed of co-factors of the matrix A. Show that if rk(A) = n - 1 then rk(B) = 1. Explain.

Answer 1

solution: Question 6 :- let us consider a natrix A = 1 1 I -1 0 J 943 de Conditions that es matrix A should be satisfy two :) A be a nan matrix w 8K (A) = -1 Here the matrix A is 3x3 order matrix square matrix satisfy Condition (i) Now we have to find ok (A) TAL - ! ! ! I- O . 11

71A1 = [-1-0] - 1[1-0] +1 [141] JAL - -1 -1 +2 IAI = 0 which is zero. so the rank of matrix A is not 3. let's take from A i, 7.11 2x2 = (1-1 – 1x1) = (-1-1) = - 2 to which is not zero. And 2.1 2x2 = (180 - 1*(-1)) = 0 + 1 = 1 * which is not zero. Hence, the rank of matrix is a that is n-1 - 3-1.& satisfy the and Condition &k(A) = n-1. Hence the matrix A which have taken is right according to the question. Now we have to prove rk (B) = 1

composed of co-factors Given that B be the of the matrix A. We know that, Cofactor (Aej) : Az = (-1) A= 1 -1 B = Co-factors of A B - - 27 0 0 0 L 1 -2 3x3 161 = lo oo a . - - 2 I 1 - 2 IBI = 1 (0-o] -(-1) [o-o]+2 [o-o] 181 - 0 + 0 + 0 B1 =0 which is zero t o Hence the rank of matrix B is not 3. let's take - 17 and 127 [from a 0 0 0 1 0 2x2 -2x2

- 1 6 : (-14 0 - (-) xe ) - 0-6 = 0 which is zero. 2)=1:3=(-1*-200) -IXO - 2 xb = (0-0) - o which is zero. Hence, the rank of matrix B is not a. Now let's take [1] w from matrix B TXT [1 ] = -1) = -1 & which is not zero. Hence the ranks of matrix B is 1 proved