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.
Question 8
Prove that |1+xyzx1+yzxy1+z|=1+x+y+z
Solution
Let L.H.S.=|1+xyzx1+yzxy1+z| Subtract third row from first row. We have, =|10−1x1+yzxy1+z| =1(1+yz+y+z−yz)−0−1(xy−x−xy) =1+y+z+x =R.H.S.
Question 9
Prove that |xpqpxqpqx|=(x−p)(x−q)(x+p+q)
Solution
Let L.H.S.=|xpqpxqpqx| Subtract second row from first row and subtract third row from second row. =|x−pp−x00x−qq−xpqx| =(x−p)(x2−qx−q2+qx)−(p−x)(−pq+px)+0 =(x−p)(x−q)(x+q)+p(x−p)(x−q) =(x−p)(x−q)(x+q+p) =(x−p)(x−q)(x+q+p) =(x−p)(x−q)(x+q+p) =R.H.S.
Question 10
Prove that |1+a1111+b1111+c|=abc(1+1a+1b+1c)
Solution
Let L.H.S.=|1+a1111+b1111+c| Subtract second row from first row and subtract third row from second row. We have, =|a−b00b−c111+c| =a(b+bc+c)+b(c)+0 =abc+bc+ac+ab =abc(1+1a+1b+1c) =R.H.S.
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