Determinants To Be Determined
A B = = ⎝ ⎜ ⎜ ⎜ ⎜ ⎛ 1 2 3 4 5 2 4 6 8 1 0 3 6 9 1 2 1 5 4 8 1 2 1 6 2 0 5 1 0 1 5 2 0 2 5 ⎠ ⎟ ⎟ ⎟ ⎟ ⎞ ⎝ ⎜ ⎜ ⎜ ⎜ ⎛ 1 − 2 3 − 4 5 − 2 3 − 4 5 − 6 3 − 4 5 − 6 7 − 4 5 − 6 7 − 8 5 − 6 7 − 8 9 ⎠ ⎟ ⎟ ⎟ ⎟ ⎞
Which of the given matrices has the greatest determinant?
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2 solutions
Basically, the same answer as Dr. Bretscher's.
By direct computation, ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ⎝ ⎜ ⎜ ⎜ ⎜ ⎛ 1 2 3 4 5 2 4 6 8 1 0 3 6 9 1 2 1 5 4 8 1 2 1 6 2 0 5 1 0 1 5 2 0 2 5 ⎠ ⎟ ⎟ ⎟ ⎟ ⎞ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ⟹ 0 and ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ⎝ ⎜ ⎜ ⎜ ⎜ ⎛ 1 − 2 3 − 4 5 − 2 3 − 4 5 − 6 3 − 4 5 − 6 7 − 4 5 − 6 7 − 8 5 − 6 7 − 8 9 ⎠ ⎟ ⎟ ⎟ ⎟ ⎞ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ⟹ 0 .
Both determinants are 0 since the rows are linearly dependent. In A row 2 is twice row 1, and in B rows 1 ,2 4, and 5 add up to zero.