The determinant Δ=| a l+a prime l prime a m+a prime m prime a n+a prime n prime b l+b prime l prime b m+b prime m prime b n+b prime n prime c l+c prime l prime c m+c prime m prime c n+c prime n prime | is equal to

Question

The determinant Δ=| a l+a prime l prime a m+a prime m prime a n+a prime n prime b l+b prime l prime b m+b prime m prime b n+b prime n prime c l+c prime l prime c m+c prime m prime c n+c prime n prime | is equal to (A) (a b c+a prime b prime c prime)(l m n+l prime m prime n prime) (B) a b c l m n+a prime b prime c prime l prime m prime n prime (C) (a2+b2+c2)(l2+m2+n2) +(a prime 2+b prime 2+c prime 2)(l prime 2+m prime 2+n prime 2) (D) 0

Tardigrade Admin · Accepted Answer

Write Δ=l Δ1+l prime Δ2, where Δ1 =|a a m+a prime m prime a n+a prime n prime b b m+b prime m prime b n+b prime n prime c c m+c prime m prime c n+c prime n prime| text and Δ2 =|a prime a m+a prime m prime a n+a prime n prime b prime b m+b prime m prime b n+b prime n prime c prime c m+c prime m prime c n+c prime n prime| In Δ1 apply C2 arrow C2-m C1, C3 arrow C3-n C1 to obtain Δ1=|a a prime m prime a prime n prime b b prime m prime b prime n prime c c prime m prime c prime n prime|=0 Similarly Δ2=0. Thus, Δ=0

Q. The determinant $Δ = ∣ ∣ a l + a^{'} l^{'} b l + b^{'} l^{'} c l + c^{'} l^{'} am + a^{'} m^{'} bm + b^{'} m^{'} c m + c^{'} m^{'} an + a^{'} n^{'} bn + b^{'} n^{'} c n + c^{'} n^{'} ∣ ∣$ is equal to

$(ab c + a^{'} b^{'} c^{'}) (l mn + l^{'} m^{'} n^{'})$

$ab c l mn + a^{'} b^{'} c^{'} l^{'} m^{'} n^{'}$

$(a^{2} + b^{2} + c^{2}) (l^{2} + m^{2} + n^{2}) + (a^{'2} + b^{'2} + c^{'2}) (l^{'2} + m^{'2} + n^{'2})$

0

Q. The determinant Δ=∣∣​al+a′l′bl+b′l′cl+c′l′​am+a′m′bm+b′m′cm+c′m′​an+a′n′bn+b′n′cn+c′n′​∣∣​ is equal to

(abc+a′b′c′)(lmn+l′m′n′)

abclmn+a′b′c′l′m′n′

(a2+b2+c2)(l2+m2+n2)+(a′2+b′2+c′2)(l′2+m′2+n′2)

0

Q. The determinant $Δ = ∣ ∣ a l + a^{'} l^{'} b l + b^{'} l^{'} c l + c^{'} l^{'} am + a^{'} m^{'} bm + b^{'} m^{'} c m + c^{'} m^{'} an + a^{'} n^{'} bn + b^{'} n^{'} c n + c^{'} n^{'} ∣ ∣$ is equal to

$(ab c + a^{'} b^{'} c^{'}) (l mn + l^{'} m^{'} n^{'})$

$ab c l mn + a^{'} b^{'} c^{'} l^{'} m^{'} n^{'}$

$(a^{2} + b^{2} + c^{2}) (l^{2} + m^{2} + n^{2}) + (a^{'2} + b^{'2} + c^{'2}) (l^{'2} + m^{'2} + n^{'2})$