Question 8,9 & 10, Exercise 2.2

Solutions of Questions 8,9 & 10 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

Prove that |1+xyzx1+yzxy1+z|=1+x+y+z

Let L.H.S.=|1+xyzx1+yzxy1+z| Subtract third row from first row. We have, =|101x1+yzxy1+z| =1(1+yz+y+zyz)01(xyxxy) =1+y+z+x =R.H.S.

Prove that |xpqpxqpqx|=(xp)(xq)(x+p+q)

Let L.H.S.=|xpqpxqpqx| Subtract second row from first row and subtract third row from second row. =|xppx00xqqxpqx| =(xp)(x2qxq2+qx)(px)(pq+px)+0 =(xp)(xq)(x+q)+p(xp)(xq) =(xp)(xq)(x+q+p) =(xp)(xq)(x+q+p) =(xp)(xq)(x+q+p) =R.H.S.

Prove that |1+a1111+b1111+c|=abc(1+1a+1b+1c)

Let L.H.S.=|1+a1111+b1111+c| Subtract second row from first row and subtract third row from second row. We have, =|ab00bc111+c| =a(b+bc+c)+b(c)+0 =abc+bc+ac+ab =abc(1+1a+1b+1c) =R.H.S.