Statement One: You are trying to evaluate ∣ ∣ ∣ ∣ ∣ ∣ 9 5 1 8 4 2 7 6 3 ∣ ∣ ∣ ∣ ∣ ∣ .
Statement Two: Then, you think that ⎝ ⎛ 9 5 1 8 4 2 7 6 3 ⎠ ⎞ = ⎝ ⎛ 5 4 3 3 5 4 4 5 3 ⎠ ⎞ + ⎝ ⎛ 4 1 − 2 5 − 1 − 2 3 1 0 ⎠ ⎞
Statement Three: ∴ ∣ ∣ ∣ ∣ ∣ ∣ 9 5 1 8 4 2 7 6 3 ∣ ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ 5 4 3 3 5 4 4 5 3 ∣ ∣ ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ ∣ 4 1 − 2 5 − 1 − 2 3 1 0 ∣ ∣ ∣ ∣ ∣ ∣
Statement Four: ⟹ ( − 3 0 ) = ( − 1 2 ) + ( − 1 4 )
Statement Five: ∴ − 3 0 = − 2 6 ?
Which of these statements is the start of the mistake in the said argument?
Notations: ∣ A ∣ is the determinant of the matrix A while ( A ) is the matrix A itself.
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It is not always true that the sum of the determinants of two matrices is the same as the determinant of the summands of the two matrices.
therefore, statement three is wrong.