Question

By using properties of determinants, show that:

Answer 1

The given left hand side determinant is,

Δ=| 1 x x 2 x 2 1 x x x 2 1 |

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

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

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

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

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

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

Expand along R 1 ,

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

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