Question

If $A = \begin{bmatrix}3&-3&4\\ 2&-3&4\\ 0&-1&1\end{bmatrix}$ , $B = (adj A) $ and $C = 5A$ then $ \frac{|C|}{|adj B|}$ is equal to

• A. 125
• B. -1
• C. 5
• D. -5

Answer 1

Answer: (A)

$A = \begin{bmatrix}3&-3&4\\ 2&-3&4\\ 0&-1&1\end{bmatrix} , |A| = 3\{(-3)+4\}3.(2) + 4(-2) = 3+6-8=1$ ...(i)
Given, $B = adj A, C = 5A$
$|C| =|5A| = 5^3|A| =5^3 =125$ (From (i))
Also, $[adj \ B |= [adj (adj \ A)| =|A|^{(3-1)^2} = |A|^4$
$\therefore \ \ \ \ \frac{\left|C\right|}{\left|adj B\right|}= \frac{125}{\left|adj \left(adj A\right)\right|}=\frac{125}{\left|A\right|^{4}} =125 \ \ \ (\because \ \ |A| =1)$