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 $P Q$ is $1$
3) Rank of $\begin{bmatrix}1 & 2 & 3 \\ 2 & 3 & 4 \\ 5 & 6 & 7\end{bmatrix}$ 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 $\begin{bmatrix}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{bmatrix}$ 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 $P Q$ is a matrix of order $1 \times 1$, so rank of matrix $P Q$ is one.
Since, the
$\begin{vmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 5 & 6 & 7 \end{vmatrix}=1(21-24)-2(14-20)+3(12-15)$
$=-3+12-9=0$ 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
$A=\begin{bmatrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{bmatrix} \Rightarrow |A|=0$
so rank of $A$ is not $3$