Question

a -b-c2a2a1. ) 2b b-c-a 2b -(a+b+c)2c2c c-a-b(ii) z2y+z+2x2-2(x y+ z)z+x+2y

Answer 1

(i)

The given left hand side determinant is,

Δ=| a−b−c 2a 2a 2b b−c−a 2b 2c 2c c−a−b |

Apply row operation R 1 → R 1 + R 2 + R 3 ,

Δ=| a−b−c+2b+2c 2a+b−c−a+2c 2a+2b+c−a−b 2b b−c−a 2b 2c 2c c−a−b | =| a+b+c a+b+c a+b+c 2b b−c−a 2b 2c 2c c−a−b | =( a+b+c )| 1 1 1 2b b−c−a 2b 2c 2c c−a−b |

Apply column operation C 1 → C 1 − C 2 ,

Δ=( a+b+c )| 1−1 1 1 2b−( b−c−a ) b−c−a 2b 2c−2c 2c c−a−b | =( a+b+c )| 0 1 1 b+c+a b−c−a 2b 0 2c c−a−b | = ( a+b+c ) 2 | 0 1 1 1 b−c−a 2b 0 2c c−a−b |

Apply column operation C 2 → C 2 − C 3 ,

Δ= ( a+b+c ) 2 | 0 1−1 1 1 b−c−a−2b 2b 0 2c−c+a+b c−a−b | = ( a+b+c ) 2 | 0 0 1 1 −( a+b+c ) 2b 0 a+b+c c−a−b | = ( a+b+c ) 3 | 0 0 1 1 −1 2b 0 1 c−a−b |

Expand along R 1 ,

Δ= ( a+b+c ) 3 [ 0−0+1 ] = ( a+b+c ) 3

Thus, the value of the left hand side of the determinant is equal to the right hand side.

(ii)

The given left hand side determinant is,

Δ=| x+y+2z x y z y+z+2x y z x z+x+2y |

Apply column operation C 1 → C 1 + C 2 + C 3 ,

Δ=| x+y+2z+x+y x y z+y+z+2x+y y+z+2x y z+x+z+x+2y x z+x+2y | =| 2( x+y+z ) x y 2( x+y+z ) y+z+2x y 2( x+y+z ) x z+x+2y | =2( x+y+z )| 1 x y 1 y+z+2x y 1 x z+x+2y |

Apply row operation R 2 → R 2 − R 3 ,

Δ=2( x+y+z )| 1 x y 1−1 y+z+2x−x y−z−x−2y 1 x z+x+2y | =2( x+y+z )| 1 x y 0 y+z+x −( x+y+z ) 1 x z+x+2y | =2 ( x+y+z ) 2 | 1 x y 0 1 −1 1 x z+x+2y |

Apply row operation R 3 → R 3 − R 1 ,

Δ=2 ( x+y+z ) 2 | 1 x y 0 1 −1 1−1 x−x z+x+2y−y | =2 ( x+y+z ) 2 | 1 x y 0 1 −1 0 0 x+y+z | =2 ( x+y+z ) 3 | 1 x y 0 1 −1 0 0 1 |

Expand along R 3 ,

Δ=2 ( x+y+z ) 3 [ 0−0+1 ] =2 ( x+y+z ) 3

Thus, the value of left hand side of the determinant is equal to the right hand side.