What is det(AB) for A = [ [1, 2], [3, 4] ] and B = [ [0, -1], [2, 3] ]?

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Multiple Choice

What is det(AB) for A = [ [1, 2], [3, 4] ] and B = [ [0, -1], [2, 3] ]?

Explanation:
The determinant of a product equals the product of the determinants, because applying two linear transformations in sequence scales volumes by the product of each transformation’s scaling factor. For 2×2 matrices, det([[a,b],[c,d]]) = ad − bc. Compute det(A): 1·4 − 2·3 = 4 − 6 = −2. Compute det(B): 0·3 − (−1)·2 = 0 + 2 = 2. Multiply the determinants: det(AB) = (−2)·2 = −4. So the determinant of AB is −4.

The determinant of a product equals the product of the determinants, because applying two linear transformations in sequence scales volumes by the product of each transformation’s scaling factor. For 2×2 matrices, det([[a,b],[c,d]]) = ad − bc.

Compute det(A): 1·4 − 2·3 = 4 − 6 = −2.

Compute det(B): 0·3 − (−1)·2 = 0 + 2 = 2.

Multiply the determinants: det(AB) = (−2)·2 = −4.

So the determinant of AB is −4.

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