Question

3 seperate questions multiple choice

Determine which of the following matrices are in RREF. ſi 0 0 27 i) 0 2 0 3 0 1 1 4 ſi 0 1 0] i) 0 1 1 0 0 0 0 1 [1 0 -1 2 ii) 0 1 07 0 o [1 0 0 2 iv) 0 1 0 1 0 0 1 0 0 0 1 iv only ii and iii ii and iv i and ii

Answer 1

echelon forma in Row - reduced An mxn matrin A is said to be row-reduced echelon matria if a The 1st non-Zero nonzero row is 1. row , element in ealh 6 in each column containing the leading 1 of some the leading A non-zero element. © there is an integer coerem) such that the first rows of A are non Zero and the remaining row Zero rows. is only are au of ith. if the leading element non-zero row occurs in the kith Column of then kickze...kky. () 203 I14 is not RREF. because st nonzero entry row is 2. ie condition @ not satisfied. in and ü - - 0 Oi, is a row reduced oool echelon matria. it satisfied all fue condition a

w) 1 0 0 - 1 0 2 o 10 is not RREF. O O because 3rd column containing leading but tuis Column Contain many nonzero element, i e a condition 6 not satisfied ۱۷ 0 0 0 2 101 0 1 0 is a row reduced echelon matrix. it satisfied all the condition. Therefore ie (ii) and (iv) is true. 3rd option is true. The giren matria A2 (32 32 -9 و بی 108 and p=5. -31 Let x = (10) o Then be the eigenrector Corresponding to d25. (A-> 7 242) X=0 3) ()() Come (*)-(3) Oy (noe 31 32 -5 -9 108 07 - -36

-) 27a-9 yao 1082-36720 ay 3x-120 » y 23re - X ( 3 Therefore, xa X= () (33) Eigen rector corresponding to tue eigen value x25 is (3) 1St option is true. so given matrin 2 The giren A= ['12 xet x be the eigen value of A. Then Characteristic ecuation is det (A-> I2x2) = 0 ror, -> 4 co 3 2-7 08, or ΟΥ (2-2) (2-7)=1220 22-32+2-1220 12-34-1020 +²57+27-10=0 or, (8-5) (7+2)0. x 25.-2 08

" eigen value of A are x=5, 22²-2, det x =( X-C.) betere eigen vector conesponding to eigen value 2,75 (A-51269) X=0 then or () (.):(0) >> -44744420 34-37²0 >> x=4 ܀ ܢ xet X; - ()():(). The eigen vector corresponding to x = 5 is (!). X₂ = ( 241 ) be the eigen rector corresponding to 722-2 then (A - (-2)72x) *220 G)) (3 3x + 4 22 ZO 12a - 3x or, 소

*2* () (W): * Therefore eigen rector corresponding to d22-2 is act us choose -47 P. - 3 [ Now because two eigen rector corres pornding to dia 5.22=22 are O. (6 respectively AP-()(-3 (5 ) (1) (3 02 Where DP (52) 2 so PtlAp=D ? Therefore pa (13 & D= 3 ) 2nd option is true.