Exercise · ZW-DETERMINANT-PROPERTIES-005 · ★★☆ Intermediate
If \(A\) and \(B\) are \(3\times3\) matrices with \(\det A=3\) and \(\det B=-2\), what is \(\det(AB)\)?
Choose one option.
Show answer
- D. \(-6\)
Explanation
The determinant is multiplicative: \(\det(AB)=\det A\,\det B=3\cdot(-2)=-6\). It is not additive: \(3+(-2)=1\) is wrong.
Review the concept: Properties of the determinant
The properties of the determinant describe how \(\det A\) changes under elementary row operations and how it behaves with the product of two matrices \(A\) and \(B\) and with the transpose \(A^T\).
\[\det(AB)=\det A\,\det B,\quad \det A^{T}=\det A\]
More practice problems: properties of the determinant
- ZW-DETERMINANT-PROPERTIES-001 Let A be a 3×3 matrix with det A=5. B is obtained by swapping two rows of A. What is det B?
- ZW-DETERMINANT-PROPERTIES-002 If det A=4 and B is obtained by multiplying a single row of A by 3, what is det B?
- ZW-DETERMINANT-PROPERTIES-003 Let A be 3×3 with det A=2. Compute det(2A).
- ZW-DETERMINANT-PROPERTIES-004 Compute det A for A=[1 2 3; 2 4 6; 7 8 10] without a full expansion.
- ZW-DETERMINANT-PROPERTIES-006 Let A be an invertible 2×2 matrix with det A=6. Compute det(3A^-1).
- ZW-DETERMINANT-PROPERTIES-007 Compute det A for A=[2 5 1; 0 -3 4; 0 0 5].
- ZW-DETERMINANT-PROPERTIES-008 If det A=-3 and B results from adding four times row 1 to row 2 of A, what is det B?
- ZW-DETERMINANT-PROPERTIES-009 Let A be a real square matrix with A^2=I. Which values can det A take?
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