True or False:
If M = ⎝ ⎛ a d g b e h c f i ⎠ ⎞ and a , b , c , d , e , f , g , h , and i are in an arithmetic progression or in a geometric progression, then det ( M ) = 0 .
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For all arithmetic sequences, let M = ⎝ ⎛ a − 4 d a − d a + 2 d a − 3 d a a + 3 d a − 2 d a + d a + 4 d ⎠ ⎞ . Then
det ( M )
= ( a − 4 d ) a ( a + 4 d ) + ( a − 3 d ) ( a + d ) ( a + 2 d ) + ( a − 2 d ) ( a − d ) ( a + 3 d ) − ( a − 2 d ) a ( a + 2 d ) − ( a − 4 d ) ( a + d ) ( a + 3 d ) − ( a − 3 d ) ( a − d ) ( a + 4 d )
= ( a 3 − 1 6 a d 2 ) + ( a 3 − 7 a d 2 − 6 d 3 ) + ( a 3 − 7 a d 2 + 6 d 3 ) − ( a 3 − 4 a d 2 ) − ( a 3 − 1 3 a d 2 − 1 2 d 3 ) − ( a 3 − 1 3 a d 2 + 1 2 d 3 )
= 0
For all geometric sequences, let M = ⎝ ⎛ a r − 4 a r − 1 a r 2 a r − 3 a a r 3 a r − 2 a r a r 4 ⎠ ⎞ . Then
det ( M )
= a r − 4 ⋅ a ⋅ a r 4 + a r − 3 ⋅ a r ⋅ a r 2 + a r − 2 ⋅ a r − 1 ⋅ a r 3 − a r − 2 ⋅ a ⋅ a r 2 − a r − 4 ⋅ a r ⋅ a r 3 − a r − 3 ⋅ a r − 1 ⋅ a r 4
= a 3 + a 3 + a 3 − a 3 − a 3 − a 3
= 0
Both arithmetic and geometric sequences result in det ( M ) = 0 , so the statement is true .
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If a , b , c , d , e , f , g , h , i are in geometric progression with common ratio r , then the second row is r 3 times the first row. Thus taking r 3 times the first row from the second row we see that d e t M = ∣ ∣ ∣ ∣ ∣ ∣ a d g b e h c f i ∣ ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ a 0 g b 0 h c 0 i ∣ ∣ ∣ ∣ ∣ ∣ = 0
If a , b , c , d , e , f , g , h , i are in arithmetic progression with common difference u , first taking the second row from the third and then the first row from the second, and then the second row from the third row again, we obtain d e t M = ∣ ∣ ∣ ∣ ∣ ∣ a d g b e h c f i ∣ ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ a d 3 u b e 3 u c f 3 u ∣ ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ a 3 u 3 u b 3 u 3 u c 3 u 3 u ∣ ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ a 3 u 0 b 3 u 0 c 3 u 0 ∣ ∣ ∣ ∣ ∣ ∣ = 0