Question

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

• A. $5$
• B. $25$
• C. $-1$
• D. $1$

Answer 1

The correct option is D $1$

We have, for a matrix order $n$, $|adj\left(adjA\right))|=|A{|}^{(n-1{)}^2}$.

Here $n=3$.

Then $|adj\left(adj\right(A\left)\right)|=|A{|}^4$.....(1).

Now for given, $A$ we have, $|A|=1(0+3)+2(3-2)=5$. [ Expanding about first column]

Now,

$\frac{|adj\left(B\right)|}{|C|}$

$=\frac{|adj\left(adj\right(A\left)\right)|}{|5A|}$ [ Using value of $B$]

$=\frac{5^4}{5^3.5}$ [ Using (1) and $|5A|=5^3|A|$]

$=1$.