Question

cIAl Prove that IA a A Multiplication of conto the second row of A an Ca C-a Ca A = as Prove that determinant of A is 0 Hint!: compare the determinants of A'along the first row and the second row. And use result of prablem 1 ap A A row of matrix A is changed into another row multiplied by constant c a1z C-a Ca AE Ca Prove that determinant ofA and determinant of A'are same. Hint!: use the result of problem 2 az .1 A row of matrix A is changed by row operalion of Gauss el iminalion. A 21 a a. az a.. a c-a a Cay a Ca A '= a

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Answer 1

Prost theorem Theo vem mafrix In nX n am be column) A ealar plied mel tipLied by e det A then

function tS-- determinant the sealar function matpicen F Auch oi)E then ( dot Caij) F and al fo J ta thok over A - dt A ) a Acalan bllonging to dtined deA det (ais) rermutahion .Where the parmmtatio an odd (n the Co hen Caye- Let theoren io when A han det A anee a

Caij)and B-(bi) ktiohaging Lat A Cafa- L be obtained from A non- 2ero a itn Acalar aii bij Then cani ond anpin arp) det A

Soi Here given AT aun au Cauu c.ain C-arz С- ari ann ani baoove theorem Then an aft arn det A an ann matrin det A given (Proved) Protf theorem above det A OL2 matrix if twoown Cor columns) be idenkical , then det A= o, identical7 nK n an схо A hene RIR ave 2A . ved)

2) Jol det A ain ain can CaM2 cau aın arr ari Cunn anr On2 Since Cach nartiulor Y th be ftuo Mor errennd term element um arj arten det (bij)t det (Cij) ) where the an a det Caij

det t the deerminank hand ide in Bince han two identicorl det A dtt A detA (Proved)