Matrices and Determinants
Properties of Determinants
Premium Question
Grade 12
Question:
For any square matrix $A$, $\det(A)$ is related to $\det(A^T)$ by:
(1) $\det(A) = -\det(A^T)$
(2) $\det(A) = \det(A^T)^{-1}$
(3) $\det(A) = \det(A^T)$
(4) $\det(A) \cdot \det(A^T) = 1$
Step-by-Step Solution
Key Concept: Recall that expanding a determinant along a row gives the same numerical value as expanding the corresponding determinant along the matching column — this is the essence of the transpose-invariance property of determinants.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)