Question

Rank of the matrix
$A=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right]$ is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

Answer: (D)

$3$

We have, $|A|=\left|\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right|$
$=1(1-4)-2(2-4)+2(4-2)$
$=-3+4+4=5\ne 0$
Thus, $A$ is non-singular matrix of order $3\times 3$.
Therefore, $r\left(A\right)=3$.