Compute the determinant of the matrix [[1,0,2],[-1,3,1],[0,4,0]].

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Prepare for the AMSOC 26-003 Module A Test. Utilize flashcards and multiple choice questions with hints and explanations. Ace your exam!

Multiple Choice

Compute the determinant of the matrix [[1,0,2],[-1,3,1],[0,4,0]].

Explanation:
Expanding along the first row uses the pattern +, -, + with cofactors. The determinant is 1 × det[[3,1],[4,0]] − 0 × det[[-1,1],[0,0]] + 2 × det[[-1,3],[0,4]]. Compute the minors: det[[3,1],[4,0]] = 3×0 − 1×4 = −4, and det[[-1,3],[0,4]] = (−1)×4 − 3×0 = −4. So the determinant = 1×(−4) − 0×(⋯) + 2×(−4) = −4 − 8 = −12. Therefore, the determinant is −12.

Expanding along the first row uses the pattern +, -, + with cofactors. The determinant is

1 × det[[3,1],[4,0]] − 0 × det[[-1,1],[0,0]] + 2 × det[[-1,3],[0,4]].

Compute the minors: det[[3,1],[4,0]] = 3×0 − 1×4 = −4, and det[[-1,3],[0,4]] = (−1)×4 − 3×0 = −4.

So the determinant = 1×(−4) − 0×(⋯) + 2×(−4) = −4 − 8 = −12.

Therefore, the determinant is −12.

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