Matrices & Determinants
Determinant
MMTS_Full_Test_20
Grade 12
Question:
Let $A$ be a $3\times 3$ matrix of non-negative real numbers such that $A\begin{bmatrix}2\\2\\2\end{bmatrix}=4\begin{bmatrix}1\\1\\1\end{bmatrix}$. Then $(\det A)_{\max}$ is
Step-by-Step Solution
Key Concept: Row sums of $A$ are all 2; maximize det by AM-GM
Each row sums to 2. By AM-GM: max det occurs when entries equal; but with row sum constraint: $(\det A)_{\max}=8$.
Correct Answer: 2