Question

If $A=\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right],B=\left(adjA\right)$ and $C=5A$ then $\frac{|C|}{|adjB|}$ is equal to

• A. $5^3$
• B. $-1$
• C. $5$
• D. $-5$

Answer 1

Answer: (A)

$5^3$

Given $A=\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right],B=\left(adjA\right)$ and $C=5A$

By property of square matrix,

$|adjA|=|A{|}^{n-1}$ where $n$ is number of row or columns of square matrix

$\frac{|C|}{|adjB|}=\frac{|5A|}{|B{|}^{3-1}}=\frac{5^3|A|}{|B{|}^2}=\frac{5^3|A|}{|AdjA{|}^2}=\frac{5^3|A|}{(|A{|}^2{)}^2}=\frac{5^3|A|}{|A{|}^4}=\frac{5^3}{|A{|}^3}$

$|A|=3\left(1\right)+3\left(2\right)+4(-2)$

$=9-8=1$

$\therefore \frac{|C|}{|adjB|}=\frac{5^3}{1}=5^3$