Question

A= 9 2 3 -9 -2 Question 4. (15 pts) Let the matrix A be the same as in Question 3 (1). Find the rank of A. (2). Find the dimensions of Nul(A) and Col(A). (3). How do the dimensions of Nul(A) and Col(A) relate to the number of columns of A?

Answer 1

Given matrix is. -3 9 - 2 1 A= 2 -6 4 8 3 -g 2 we betore solving any question, first reduce the matrix A in its echelon form. (ret) row Applying row operations. R. → R: + 2 / 3 Ri R3 Rz7R3 > 3 9 - 2 7 38 8 3 o 9 RI ģ Ri R2 → colw po -3 2/3 -773 : 3878 -4 > CS Scanned with CamScanner

R = RI - - 23 R2 - 3 블 o قام ها o 0 R3 → R3 + 4 R2 1 - 3 o - 11 / 2 1914 0 0 0 28 R3 I R3 28 3 o -1/ 1914 0 o o RI R + 1/2 R3 R2 = R₂ - 12 / 23 ņoo оо оо Now it us the rret of A. CS Scanned with Cam Scanner

O А. rank of number of in rret of A nonzero rows in rows There are three nonzero met (A) Therefore, rank of А = 3. dimension of null (A) or Nul (A) is equal to number of rows in rret (A) which have all entriez o. There no any row that have all entries SO nul (A) 2 = 0 1 dimension of col (A) number of leading in ret (A) There 3 (three) leading 1 in so ColCA) - 3 are met (A) The dimension of NUILA) and col (A) always less than or equal to the number of Columns of A. CS Scanned with CamScanner