Let A be a $3 \times 3$ 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 ____.

JEE Main · 2024 · 04 Apr 2024 · Shift 1 · Matrices

Let A be a

3 \times 3

matrix of non-negative real elements such that

A 111 = 3111

. Then the maximum value of

det (A)

is ____.

Answer: 27

A 111 = 333

means every row of the non-negative matrix sums to

3

det A

is maximized when the matrix is as "diagonal" as possible; with each row summing to 3 the maximum is achieved at

A = 3 I

det A_{m a x} = 3 \cdot 3 \cdot 3 = 27.

Answer: 27.

JEE Main · 2024 · 04 Apr 2024 · Shift 1 · Integer answer