Question

7. (16 pts.) We are given a 4 x 4 matrix A, with rows r1,..., r4. The matrix B is obtained from the rows of A as described below: 12 A= B 4r + ri +2r3 2r2 + 4r r3 +r2 3r1 13 TA How are the determinants of A and B related?

Answer 1

let an 4 A ай az ala аз а4 а32 33 44 ам a43 44 . ant we write А=. where ті 8, first row of A zond row of A * * Тэ 1, 2 аш 2 аз 14 т.= а21 а22 223 224 *2 = аз аз 3 4 ТА ЗА 442 43 44 3rd row of A You od А. 4 Let determinant of A = det CA)=k Бg Gra scalar,

k = det(A)= ra ชน та - - interchange & and fa, тъ 1. Үз Te -с АТА multiply first row by 4. 덧 4. multiply by 2 by and son . 1,1. А 2 2 ana mulpa 2 va Ra. AYA 272 2Yэ ата

1) 4:12 ANA+28+7 2т, тә R= Te Ist row + 2nd row + 4 ++ +ою,

--tait 1984 +84 +273 282 282+28 Rig = R₂ + R2. LARA Ar1 -- to Å 1 A8A+% +2% R2 = R₂ + Rp 282 tari 222 +273 4r :- 16 artrat 283 282 +401 --* ģ AVA+&+ 28 282+ 481 282+28 } +8B 301 - =4 determinant of 3 , 8 3 therefore determinant of B=-24 odeterminant of a