Question

13. 0.19/1.33 points Previous Answers LARLINALG8 4.6.041. Use the fact that matrices A and B are row-equivalent. [ 1 2 1 0 0 2 5 1 1 0 3 7 2 2 - 2 5 11 4 -3 8 1 0 3 0 -4 0 1 -1 0 2 Loo 01 - Loo 0 0 0 (a) Find the rank and nullity of A. rank nullity (b) Find a basis for the nullspace of A.

Answer 1

Given en that A1 that A and B are now equivalent, where T 2 0 0 10 3 0-4 2 5 10 01-02 and B = 1 13 7 2 2 - 2 looo 1-2 0 0 15 11 0 0 4 -3 0 8 Mote (i) Rank of a Rank of a matrún = Number of non-zero rows in the reduced matrix. (ii) Wallity of a matrin - Numb mabrin a Number in the of free variable reduced matrix J.. Now it is is given given that that is B in the reduced matrin of A. (a) rank (A) = 3 [ there are there are three non-zoru rows in B! three non- Now the e system bystem Bx=0 Bx=0 can be written an x + 303 -ANG = 0 2 - 2 +225 20P - 2*5 = 0 ne z and who are free rasiables: i nullity of (A) = 2. nullity = Since x and Me are free variable so from we get as follows by taking a cons=d. x = aze+ad My = erad x = 2d. i will space of A i I-3etad 7 c-2d 20 edER?

le nult space of a - + 2 id EIRE oo -span f TA " Wallity of A in 2 Lo ? I of Ain 2 and it is spanded by two vectors so this vectors are basis for the null i. Basis for the null space of U is CO- 3 A is they and form the first a basis three row of B are for the row space ince rank of non-zero, so of A. Basis for the row Moace of his s [ 20-4]? We apply column oporation on B, we got C + 2 - 2 + 20 0 0 3 + 19 - 3G+ Doo of A is 3 and non-zero, so the they 1st three columm of the reduced matris form a basis for the entumn space Since rank of B are of A. i. Basis for of A in column space the I lolos