Question

+42x2x4 2x 210. )2x x+42x -(5x+4) (4-x)2x 2x x+y+k y+ k

Answer 1

(i)

The given left hand side determinant is,

Δ=| x+4 2x 2x 2x x+4 2x 2x 2x x+4 |

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

Δ=| x+4+2x+2x 2x+x+4+2x 2x+2x+x+4 2x x+4 2x 2x 2x x+4 | =| 5x+4 5x+4 5x+4 2x x+4 2x 2x 2x x+4 | =( 5x+4 )| 1 1 1 2x x+4 2x 2x 2x x+4 |

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

Δ=( 5x+4 )| 1−1 1 1 2x−x−4 x+4 2x 2x−2x 2x x+4 | =( 5x+4 )| 0 1 1 x−4 x+4 2x 0 2x x+4 | =( 5x+4 )( x−4 )| 0 1 1 1 x+4 2x 0 2x x+4 |

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

Δ=( 5x+4 )( x−4 )| 0 1−1 1 1 x+4−2x 2x 0 2x−x−4 x+4 | =( 5x+4 )( x−4 )| 0 0 1 1 −( x−4 ) 2x 0 x−4 x+4 | =( 5x+4 )( x−4 )( x−4 )| 0 0 1 1 −1 2x 0 1 x+4 |

Expand along R 1 ,

Δ=( 5x+4 ) ( x−4 ) 2 [ 0−0+1 ] =( 5x+4 ) ( x−4 ) 2

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

(ii)

The given left hand side determinant is,

Δ=| y+k y y y y+k y y y y+k |

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

Δ=| y+k+y+y y+y+k+y y+y+y+k y y+k y y y y+k | =| 3y+k 3y+k 3y+k y y+k y y y y+k | =( 3y+k )| 1 1 1 y y+k y y y y+k |

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

Δ=( 3y+k )| 1−1 1 1 y−y−k y+k y y−y y y+k | =( 3y+k )| 0 1 1 −k y+k y 0 y y+k | =k( 3y+k )| 0 1 1 −1 y+k y 0 y y+k |

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

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

Expand along R 1 ,

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

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