Question

A value of $b$ for which the rank of the matrix
$A=\left[\begin{array}{cccc}1& 1& -1& 0\\ 4& 4& -3& 1\\ b& 2& 2& 2\\ 9& 9& b& 3\end{array}\right]$ is $3$, is

• A. -2
• B. -4
• C. -6
• D. 3

Answer 1

Answer: (C)

Given,
$A=\left[\begin{array}{cccc}1& 1& -1& 0\\ 4& 4& -3& 1\\ b& 2& 2& 2\\ 9& 9& b& 3<br/><br/>\end{array}\right]$
For rank to be 3, there must exist 3 non zero row.
Now, applying $R_2\to R_2-4R_1;R_3\to R_3-2R_1$
$=\left[\begin{array}{cccc}1& 1& -1& 0\\ 0& 0& 1& 1\\ b-2& 0& 4& 2\\ 9& 9& b& 3<br/><br/>\end{array}\right]$
Applying $R_4\to R_4-9R_1$
$=\left[\begin{array}{cccc}1& 1& -1& 0\\ 0& 0& 1& 1\\ b-2& 0& 4& 2\\ 0& 0& b+9& 3<br/><br/>\end{array}\right]$
Again, applying $R_4\to R_4-3R_2$
$A=\left[\begin{array}{cccc}1& 1& -1& 0\\ 0& 0& 1& 1\\ b-2& 0& 4& 2\\ 0& 0& b+6& 0<br/><br/>\end{array}\right]$
If rank $=3$, then
Last row must have all elements $0$
$\therefore b+6=0$
$\Rightarrow b=-6$