Question

Let $A=\begin{bmatrix} 1 & 1 & 1 \\ 1 & -1 & 0 \\ 0 & 1 & -1 \end{bmatrix}$ , $A_{1}$ 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)

$\left|A\right|=1\left(1 - 0\right)-1\left(- 1 - 1\right)=3$
$\left|A_{1}\right|=\left(\left|A\right|\right)^{2}\left(b y p r o p e r t y\right)$
$\left|A_{2}\right|=\left|A_{1}\right|^{2}=\left|A\right|^{4}$
$\left|A_{3}\right|=\left|A_{2}\right|^{2}=\left|A\right|^{8}$
Similarly, $\left|A_{10}\right|=\left|A\right|^{2^{10}}=3^{2^{10}}=3^{1 024}$