Matrices and Determinants
Trace of a Matrix
Premium Question
Grade 12
Question:
For square matrices $A$ and $B$ of the same order, $\text{trace}(AB)$ equals:
(1) $\text{trace}(BA)$
(2) $\text{trace}(A) \cdot \text{trace}(B)$
(3) $\text{trace}(A) + \text{trace}(B)$
(4) $\text{trace}(A^{-1}B)$
Step-by-Step Solution
Key Concept: Write out the $(i, i)$ diagonal entry of $AB$ as $\sum_k a_{ik}b_{ki}$, sum over $i$ to get $\text{trace}(AB)$, and compare this double sum with the one you would get for $\text{trace}(BA)$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)