Question

Let $A=\left[\begin{array}{ccc}1& 1& 1\\ 1& -1& 0\\ 0& 1& -1\end{array}\right]$ , $A_1$ be a matrix formed by the cofactors of the elements of the matrix $A$ and $A_2$ be a matrix formed by the cofactors of the elements of matrix $A_1.$ Similarly, if $A_{10}$ be a matrix formed by the cofactors of the elements of matrix $A_9$ , then the value of $\left|A_{10}\right|$ is

• A. $3^{10}$
• B. $3^{20}$
• C. $9$
• D. $3^{1024}$

Answer 1

Answer: (D)

$\left|A\right|=1\left(1-0\right)-1\left(-1-1\right)=3$
$\left|A_1\right|={\left(\left|A\right|\right)}^2\left(byproperty\right)$
$\left|A_2\right|={\left|A_1\right|}^2={\left|A\right|}^4$
$\left|A_3\right|={\left|A_2\right|}^2={\left|A\right|}^8$
Similarly, $\left|A_{10}\right|={\left|A\right|}^{2^{10}}=3^{2^{10}}=3^{1024}$