Compute the determinant of the matrix A = [ [1,2,3], [4,5,6], [7,8,9] ].

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Multiple Choice

Compute the determinant of the matrix A = [ [1,2,3], [4,5,6], [7,8,9] ].

The determinant is zero because the rows are linearly dependent. The difference between consecutive rows is the same: Row2 minus Row1 equals [3,3,3] and Row3 minus Row2 equals [3,3,3]. This gives Row3 = 2·Row2 − Row1, so a nontrivial linear combination of the rows equals zero. When rows (or columns) are linearly dependent, the determinant vanishes. Quick check by expansion: det A = 1(5·9 − 6·8) − 2(4·9 − 6·7) + 3(4·8 − 5·7) = 1(−3) − 2(−6) + 3(−3) = −3 + 12 − 9 = 0.

The determinant is zero because the rows are linearly dependent. The difference between consecutive rows is the same: Row2 minus Row1 equals [3,3,3] and Row3 minus Row2 equals [3,3,3]. This gives Row3 = 2·Row2 − Row1, so a nontrivial linear combination of the rows equals zero. When rows (or columns) are linearly dependent, the determinant vanishes.

Quick check by expansion: det A = 1(5·9 − 6·8) − 2(4·9 − 6·7) + 3(4·8 − 5·7) = 1(−3) − 2(−6) + 3(−3) = −3 + 12 − 9 = 0.

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