What is the value of \begin{vmatrix}a_{21}&a_{31}&a_{11}\\ a_{23}&a_{33}&a_{13}\\ a_{22}&a_{32}&a_{12}\end{vmatrix}?
Consider the following for the next three (03) items that follow : Let \Delta be the determinant of a matrix A, where A=\begin{pmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{pmatrix} and C_{11}, C_{12}, C_{13} be the cofactors of a_{11}, a_{12}, a_{13} respectively.
- A. 0
- B. 1
- C. \Delta
- D. -\Delta ✓
Correct Answer: D. -\Delta
Explanation
Taking the transpose of the given determinant does not change its value. Moving the bottom row to the top requires two row swaps, which maintains the sign. Finally, swapping the second and third columns multiplies the determinant by -1. The result is -\Delta.
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