Which of the following is false? 1. If \(A\) is a skew symmetric matrix of order \(5 \times 5\), then the…
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 non-zero column matrix and \(Q\) is a non-zero row matrix, then the rank of \(P Q\) is 1
3. Rank of \(\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 3 & 4 \\ 5 & 6 & 7\end{array}\right]\) 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 \(\operatorname{rank}\) of \(\left[\begin{array}{lll}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right]\) is
1
2
3
4
Solution
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
\(\left|\begin{array}{lll}
1 & 2 & 3 \\
2 & 3 & 4 \\
5 & 6 & 7
\end{array}\right|=1(21-24)-2(14-20)+3(12-15)\)
\(=-3+12-9=0\) 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
\(A=\left[\begin{array}{lll}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right] \Rightarrow|A|=0\), so rank of \(A\) is not 3.