Question

If $A$ is a $3\times 3$ matrix such that $|A|=5,$ then the value of $|adj\left(4A\right)|$ is

• A. ${16}^3\times 5^2$
• B. ${12}^3\times 5^2$
• C. ${16}^3\times 6^2$
• D. ${14}^3\times 5^2$

Answer 1

The correct option is A ${16}^3\times 5^2$

We know $adj\left(kA\right)=k^{n-1}\left(adj\right(A\left)\right),k\in R$
$\Rightarrow |adj\left(4A\right)|=|4^{3-1}\cdot adjA|$
$\Rightarrow |adj\left(4A\right)|=|16adj\left(A\right)|$
$={16}^3\cdot |adjA|$
$={16}^3\cdot |A{|}^{n-1}$
$={16}^3\cdot 5^{3-1}$
$={16}^3\times 5^2$

Alternatively,
$adj\left(kA\right)=|kA{|}^{n-1}$
$\Rightarrow adj\left(4A\right)=|4A{|}^2$
$\Rightarrow adj\left(4A\right)=(4^3{)}^2|A{|}^2$
$\Rightarrow adj\left(4A\right)=(4^2{)}^3|A{|}^2=(4^2{)}^3\times 5^2$