# Let $A$ be a nonsingular square matrix of order $3 \times 3.$ Then $| adj \;A|$ is equal to:

$\quad (A)\; |\;A\;| \quad (B)\; |\;A\;|^{2} \quad (C)\; |\;A\;|^{3} \quad (D)\; 3|\;A\;|$

Toolbox:
• A matrix is said to be singular if $|A| \neq 0$.
We know (adj A)A=|A|I=$\begin{vmatrix}|A| & 0 & 0\\0 & |A| & 0\\0 & 0 &|A|\end{vmatrix}$

$\;\;\;|(adj\;A)A|=\begin{vmatrix}|A| & 0 & 0\\0 & |A| & 0\\0 & 0 &|A|\end{vmatrix}$

Taking |A| as a common factor of $R_1,R_2$ and $R_3$.

$|adj\;A|\times |A|=|A|^3\begin{vmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 &1\end{vmatrix}$

$|adj \;A|.|A|=|A^3|I$

$|adj \;A|=|A^2|I$

Hence $|adj\; A|=|A|^2$.

Hence B is the correct answer.

edited Feb 28, 2013