Question

If $A$ is a square matrix of order $3$ and $||adj\left(A\right)|\cdot |A|\cdot A|=|A{|}^{\lambda },$ then the value of $\lambda$ is

• A. $10$
• B. $15$
• C. $5$
• D. $4$

Answer 1

The correct option is A $10$

For a $n\times n$ matrix,
$||adj\left(A\right)|\cdot |A|\cdot A|$
$=||A{|}^{n-1}\cdot |A|\cdot A|$
$=||A{|}^n\cdot A|$
$=(|A{|}^n{)}^n\cdot |A|$
$=|A{|}^{n^2}\cdot |A|=|A{|}^{n^2+1}$

Here, $n=3$
So, $||adj\left(A\right)|\cdot |A|\cdot A|$
$=|A{|}^{3^2+1}=|A{|}^{\lambda }$
$\therefore \lambda =10$