Question

If $A=\left[\begin{array}{ccc}1& -1& 1\\ 0& 2& -3\\ 2& 1& 0\end{array}\right],B=\left(adjA\right)$ and $C=5A$, then $\frac{\left|adjB\right|}{\left|C\right|}$ is equal to

• A. $5$
• B. $25$
• C. $-1$
• D. $1$

Answer 1

The correct option is A $5$

We have

$\begin{array}{c}A=\left[\begin{array}{ccc}1& -1& 1\\ 0& 2& -3\\ 2& 1& 0\end{array}\right]\\ B=\left(adjA\right)\\ And,\\ C=5A\\ Now\\ \frac{\left|adjB\right|}{\left|C\right|}=?\\ \left|A\right|=1\left(0+3\right)+1\left(6\right)+1\left(-4\right)\\ \left|A\right|=5\\ \frac{\left|adj\left(adjA\right)\right|}{\left|5A\right|}=\frac{{\left|A\right|}^{{\left(3-1\right)}^2}}{125\left|A\right|}\\ =\frac{{\left|A\right|}^4}{125|A|}\\ =\frac{|A|}{125}=1\end{array}$

Hence,

The option $\left(A\right)$ is the correct answer.