Let A=[a_ij] be a 33 matrix, where a_ij=cases(-1)^j-1&if ijcases.

Question

Let $A=[a_{ij}]$ be a $3\times3$ matrix, where $a_{ij}=\begin{cases}(-1)^{j-1}&\text{if }i<j\2&\text{if }i=j\(-1)^{i+j}&\text{if }i>j\end{cases}$. Then $\det\big(3\operatorname{Adj}(2A^{-1})\big)$ is equal to \_\_\_\_\_\_.

Answer: 108

Accepted Answer

$A=\begin{pmatrix}2&-1&1\-1&2&-1\1&-1&2\end{pmatrix}$, and $|A|=4$.

$$\big|3\operatorname{Adj}(2A^{-1})\big|=3^3\cdot\big|2A^{-1}\big|^2=27\cdot2^6\cdot|A^{-1}|^2=27\cdot64\cdot\frac1{16}=27\times4=108.$$

Solutions · 1