Let A be a square matrix of order 3 such that

Solution:

Given:

$A$ be a square matrix of order 3

$|A|=11$

$B$ be the matrix of cofactors of elements of $A$

Since, $B$ be the matrix of cofactors of elements of $A$

$\Rightarrow B=(\operatorname{adj} A)^{T}$

$\Rightarrow|B|=\left|(\operatorname{adj} A)^{T}\right|$

$\Rightarrow|B|=|\operatorname{adj} A|$

$\Rightarrow|B|^{2}=|\operatorname{adj} A|^{2}$$\quad\left(\because\left|A^{T}\right|=|A|\right)$

As we know,

$|\operatorname{adj} A|=|A|^{n-1}$, where $n$ is the order of $A$

$\Rightarrow|B|=|A|^{n-1}$

$\Rightarrow|B|=|A|^{3-1} \quad(\because$ Order of $A$ is 3$)$

$\Rightarrow|B|=|A|^{2}$

$\Rightarrow|B|=(11)^{2} \quad(\because|A|=11)$

$\Rightarrow|B|=121$

$\Rightarrow|B|^{2}=(121)^{2}$

$\Rightarrow|B|^{2}=14641$

Hence, $|B|^{2}=\underline{14641}$.