Question

Which of the following is false?
1) If $A$ is a skew symmetric matrix of order $5\times 5$, then the rank of $A$ is less than $5$
2) If $P$ is a nonzero column matrix and $Q$ is a non-zero row matrix, then the rank of $PQ$ is $1$
3) Rank of $\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 5& 6& 7\end{array}\right]$ is $2$
4) If the lines $a_{r}x+b_{r}y+c_r=0(r=1,2,3)$ are distinct and intersect at a point, then rank of $\left[\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right]$ is $3$

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (D)

The skew-symmetric matrix of order $5\times 5$, has rank less than $5$ because determinant of odd ordered skew symmetric matrix is zero.
If $P$ is a non-zero column matrix and $Q$ is a non-zero row matrix, then $PQ$ is a matrix of order $1\times 1$, so rank of matrix $PQ$ is one.
Since, the
$\left|\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 5& 6& 7\end{array}\right|=1(21-24)-2(14-20)+3(12-15)$
$=-3+12-9=0$ and there is no cofactor of any elements is zero, so rank is 2 .
If the lines $a_{r}x+b_{r}y+c_r=0,(r=1,2,3)$ are distinct and intersect at a point, then matrix
$A=\left[\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right]\Rightarrow |A|=0$
so rank of $A$ is not $3$