Matrices & Determinants
Maximum Determinant — Non-Negative Elements with Row-Sum Constraint
nta_pyq_2024_apr
Grade 12
Question:
Let $A$ be a $3\times3$ matrix of non-negative real elements such that $A\begin{bmatrix}1\\1\\1\end{bmatrix}=3\begin{bmatrix}1\\1\\1\end{bmatrix}$. Then the maximum value of $\det(A)$ is
Step-by-Step Solution
Key Concept: The condition means each row of $A$ sums to 3: $a_1+a_2+a_3=b_1+b_2+b_3=c_1+c_2+c_3=3$. Maximize $\det(A)=a_1b_2c_3+\cdots$ subject to row sums $=3$ and all entries $\geq0$.
$\det(A)_{\max}=27$ at $a_1=b_2=c_3=3$.
Correct Answer: 27