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Home  >>  CBSE XII  >>  Math  >>  Determinants
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Let $A$ be a nonsingular square matrix of order $3 \times 3.$ Then $| adj \;A| $ is equal to:

\[ \quad (A)\; |\;A\;| \quad (B)\; |\;A\;|^{2} \quad (C)\; |\;A\;|^{3} \quad (D)\; 3|\;A\;| \]
Can you answer this question?
 
 

1 Answer

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Toolbox:
  • A matrix is said to be singular if $|A| \neq 0$.
  • (Adj A)A=|A|I.
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.

 

answered Feb 26, 2013 by sreemathi.v
edited Feb 28, 2013 by vijayalakshmi_ramakrishnans
 

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