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1)

If x,y are any two  non-zero real numbers , aij=xi+yj,A=(aij)nxn and P.Q are two n x n  matrices such that A= xP+ yQ,  then


A) P is singular and Q is non-singular

B) P+Q is symmetric and P-Q is skew symmetric

C) Both P+Q and P-Q are singular

D) Both P+Q and P-Q are non-singular

Answer:

Option B

Explanation:

We have  , aij=xj+yi

A=(ai)nxn

A=[x+y2x+y3x+y.....nx+yx+2y2x+2y3x+2y......nx+2yx+3y............x+ny.......nx+ny]n×n

 A=[123......n123.....n1............123.....n]+y[111.....12....23.......3n....n]

A=xP+yQ

where

P=[123n123n12.........123n]andQ[111....1222....23.......3....n..n...n]

    (P+Q)   is symmetric and (P-Q) is skew symmetric