Question

A value of $b$ for which the rank of the matrix
$A = \begin{bmatrix}1&1&-1&0\\ 4&4&-3&1\\ b&2&2&2\\ 9&9&b&3\end{bmatrix} $ is $3$, is

• A. -2
• B. -4
• C. -6
• D. 3

Answer 1

Answer: (C)

Given,
$A=\begin{bmatrix}1 & 1 & -1 & 0 \\4 & 4 & -3 & 1 \\b & 2 & 2 & 2 \\9 & 9 & b & 3
\end{bmatrix}$
For rank to be 3, there must exist 3 non zero row.
Now, applying $R_{2} \rightarrow R_{2}-4 R_{1} ; R_{3} \rightarrow R_{3}-2 R_{1}$
$=\begin{bmatrix} 1 & 1 & -1 & 0 \\0 & 0 & 1 & 1 \\b-2 & 0 & 4 & 2 \\9 & 9 & b & 3
\end{bmatrix}$
Applying $ R_{4} \rightarrow R_{4}-9 R_{1} $
$=\begin{bmatrix} 1 & 1 & -1 & 0 \\0 & 0 & 1 & 1 \\b-2 & 0 & 4 & 2 \\0 & 0 & b+9 & 3
\end{bmatrix}$
Again, applying $R_{4} \rightarrow R_{4}-3 R_{2}$
$ A=\begin{bmatrix} 1 & 1 & -1 & 0 \\ 0 & 0 & 1 & 1 \\ b-2 & 0 & 4 & 2 \\ 0 & 0 & b+6 & 0
\end{bmatrix}$
If rank $ = 3$, then
Last row must have all elements $0$
$\therefore b + 6 = 0$
$\Rightarrow b = -6$