Exercise · ZW-DETERMINANT-PROPERTIES-010 · ★★★ Advanced
Let \(A\) be a \(3\times3\) matrix with \(A^T=-A\). What is \(\det A\)?
Choose one option.
Show answer
- D. \(0\)
Explanation
\(\det A=\det A^T=\det(-A)=(-1)^3\det A=-\det A\), so \(2\det A=0\) and \(\det A=0\). This only works for odd order.
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-005 If A and B are 3×3 matrices with det A=3 and det B=-2, what is det(AB)?
- 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?