Question

Let $A$ be a nonsingular square matrix of order $3\times 3$. Then $|adjA|$ is equal to

• A. $|A|$
• B. ${|A|}^2$
• C. ${|A|}^3$
• D. $3|A|$

Answer 1

Answer: (B)

${|A|}^2$

We know that

$AadjA=|A|I_n$

$\Rightarrow |AadjA|=\left||A|I_n\right|$

$\Rightarrow |A||adjA|={|A|}^n\left|I_n\right|$ ($\because |AB|=|A||B|;|kA|=k^n|A|$)

$|A||adjA|={|A|}^n\left|I_n\right|$ (Determinant of identity matrix is $1$)

Dividing by $|A|$, we get

$\Rightarrow |adjA|={|A|}^{n-1}$ (Since, $A$ is non-singular i.e.$|A|\ne 0$)

So, if $A$ is a square matrix of order $3$,

$|adjA|=|A{|}^2$