Let A be a 3 3 matrix of non-negative real elements such that A[[|gathered 1 1 1 gathered]…

Question

Let $A$ be a $3	imes 3$ matrix of non-negative real elements such that $A[[|\begin{gathered} 1 \ 1 \ 1 \end{gathered}]]|=3[[|\begin{gathered} 1 \ 1 \ 1 \end{gathered}]]|$. Then the maximum value of $\det (A)$ is \_\_\_\_\_\_\_\_\_.

Answer: 27

Accepted Answer

Let $A=[[|\begin{gathered} {a}_{11} \quad  {a}_{12} \quad  {a}_{13} \ {a}_{21} \quad  {a}_{22} \quad  {a}_{23} \ {a}_{31} \quad  {a}_{32} \quad  {a}_{33} \end{gathered}]]|$

Now

$A[[|\begin{gathered} 1 \ 1 \ 1 \end{gathered}]]|=3[[|\begin{gathered} 1 \ 1 \ 1 \end{gathered}]]|$

$[[|\begin{gathered} {a}_{11} \quad  {a}_{12} \quad  {a}_{13} \ {a}_{21} \quad  {a}_{22} \quad  {a}_{23} \ {a}_{31} \quad  {a}_{32} \quad  {a}_{33} \end{gathered}]]|[[|\begin{gathered} 1 \ 1 \ 1 \end{gathered}]]|=[[|\begin{gathered} 3 \ 3 \ 3 \end{gathered}]]|$

$\begin{gathered} {a}_{11}+{a}_{12}+{a}_{13}=3 \ {a}_{21}+{a}_{22}+{a}_{23}=3 \ {a}_{31}+{a}_{32}+{a}_{33}=3 \end{gathered}$

Now for maximum value of $\det (A)={a}_{ij} \begin{gathered} 0 i
e j \ 3 i=j \end{gathered}$

$	herefore |A|=27$

Solutions · 1